473G: CS Combinatorial Algorithms, Fall 2005
Homework 0
Due Thursday, September 1, 2005, at the beginning of class (12:30pm CDT)
Name:
Net ID:
Alias:
I understand the Homework Instructions and FAQ.
Neatly print your full name, your NetID, and an alias of your choice in the boxes above.
Grades will be listed on the course web site by alias. Please write the same alias on every
homework and exam! For privacy reasons, your alias should not resemble your name or
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alias, your grades will not be listed. Never give us your Social Security number!
Read the Homework Instructions and FAQ on the course web page, and then check the box
above. This page describes what we expect in your homework solutionsstart each numbered
problem on a new sheet of paper, write your name and NetID on every page, dont turn in
source code, analyze and prove everything, use good English and good logic, and so onas
well as policies on grading standards, regrading, and plagiarism. See especially the course
policies regarding the magic phrases I dont know and and so on. If you have
any questions, post them to the course newsgroup or ask during lecture.
Dont forget to submit this cover sheet with the rest of your homework solutions.
This homework tests your familiarity with prerequisite materialbig-Oh notation, elementary algorithms and data structures, recurrences, discrete probability, and most importantly,
inductionto help you identify gaps in your knowledge. You are responsible for lling
those gaps on your own. Chapters 110 of CLRS should be sucient review, but you may
also want consult your discrete mathematics and data structures textbooks.
Every homework will have ve required problems. Most homeworks will also include one
extra-credit problem and several practice (no-credit) problems. Each numbered problem is
worth 10 points.
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CS 473G
Homework 0 (due September 1, 2005)
Fall 2005
1. Solve the following recurrences. State tight asymptotic bounds for each function in the form
(f (n)) for some recognizable function f (n). You do not need to turn in proofs (in fact, please
dont turn in proofs), but you should do them anyway, just for practice. Assume reasonable
but nontrivial base cases. If your solution requires specic base cases, state them!
(a) A(n) = 2A(n/4) +
(b) B (n) =
max
n
B (k ) + B (n k ) + n
n/3<k<2n/3
(c) C (n) = 3C (n/3) + n/ lg n
(d) D(n) = 3D(n 1) 3D(n 2) + D(n 3)
(e) E (n) =
E (n 1)
3E (n 2)
[Hint: This is easy!]
(f) F (n) = F (n 2) + 2/n
(g) G(n) = 2G
(n + 3)/4 5n/ lg n + 6 lg lg n + 7 8 n 9 lg10 n/ lg lg n + 11lg n 12
(h) H (n) = 4H (n/2) 4H (n/4) + 1
[Hint: Careful!]
(i) I (n) = I (n/2) + I (n/4) + I (n/8) + I (n/12) + I (n/24) + n
(j) J (n) = 2 n J ( n) + n
[Hint: First solve the secondary recurrence j (n) = 1 + j ( n).]
2. Penn and Teller agree to play the following game. Penn shues a standard deck1 of playing
cards so that every permutation is equally likely. Then Teller draws cards from the deck, one
at a time without replacement, until he draws the three of clubs (3), at which point the
remaining undrawn cards instantly burst into ames and the game is over.
The rst time Teller draws a card from the deck, he gives it to Penn. From then on, until
the game ends, whenever Teller draws a card whose value is smaller than the previous card
he gave to Penn, he gives the new card to Penn. To make the rules unambiguous, they agree
on the numerical values A = 1, J = 11, Q = 12, and K = 13.
(a) What is the expected number of cards that Teller draws?
(b) What is the expected maximum value among the cards Teller gives to Penn?
(c) What is the expected minimum value among the cards Teller gives to Penn?
(d) What is the expected number of cards that Teller gives to Penn?
Full credit will be given only for exact answers (with correct proofs, of course).
1
In a standard deck of 52 cards, each card has a suit in the set {, , , } and a value in the set
{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K }, and every possible suit-value pair appears in the deck exactly once. Penn and
Teller normally use exploding razor-sharp ninja throwing cards for this trick.
1
CS 473G
Homework 0 (due September 1, 2005)
Fall 2005
3. A rolling die maze is a puzzle involving a standard six-sided die2 and a grid of squares. You
should imagine the grid lying on top of a table; the die always rests on and exactly covers
one square. In a single step, you can roll the die 90 degrees around one of its bottom edges,
moving it to an adjacent square one step north, south, east, or west.
Rolling a die.
Some squares in the grid may be blocked ; the die can never rest on a blocked square. Other
squares may be labeled with a number; whenever the die rests on a labeled square, the number
of pips on the top face of the die must equal the label. Squares that are neither labeled nor
marked are free. You may not roll the die o the edges of the grid. A rolling die maze is
solvable if it is possible to place a die on the lower left square and roll it to the upper right
square under these constraints.
For example, here are two rolling die mazes. Black squares are blocked. The maze on the left
can be solved by placing the die on the lower left square with 1 pip on the top face, and then
rolling it north, then north, then east, then east. The maze on the right is not solvable.
Two rolling die mazes. Only the maze on the left is solvable.
(a) Suppose the input is a two-dimensional array L[1 .. n][1 .. n], where each entry L[i][j ]
stores the label of the square in the ith row and j th column, where 0 means the square
is free and 1 means the square is blocked. Describe and analyze a polynomial-time
algorithm to determine whether the given rolling die maze is solvable.
(b) Now suppose the maze is specied implicitly by a list of labeled and blocked squares.
Specically, suppose the input consists of an integer M , specifying the height and width
of the maze, and an array S [1 .. n], where each entry S [i] is a triple (x, y, L) indicating
that square (x, y ) has label L. As in the explicit encoding, label 1 indicates that the
square is blocked; free squares are not listed in S at all. Describe and analyze an ecient
algorithm to determine whether the given rolling die maze is solvable. For full credit,
the running time of your algorithm should be polynomial in the input size n.
[Hint: You have some freedom in how to place the initial die. There are rolling die mazes
that can only be solved if the initial position is chosen correctly.]
2
A standard die is a cube, where each side is labeled with a dierent number of dots, called pips, between 1 and 6.
The labeling is chosen so that any pair of opposite sides has a total of 7 pips.
2
CS 473G
Homework 0 (due September 1, 2005)
Fall 2005
4. Whenever groups of pigeons gather, they instinctively establish a pecking order. For any pair
of pigeons, one pigeon always pecks the other, driving it away from food or potential mates.
The same pair of pigeons will always chooses the same pecking order, even after years of
separation, no matter what other pigeons are around. (Like most things, revenge is a foreign
concept to pigeons.) Surprisingly, the overall pecking order in a set of pigeons can contain
cyclesfor example, pigeon A pecks pigeon B, which pecks pigeon C, which pecks pigeon A.
Prove that any set of pigeons can be arranged in a row so that every pigeon pecks the pigeon
immediately to its right.
5. Scientists have recently discovered a planet, tentatively named Ygdrasil, which is inhabited
by a bizarre species called vodes. All vodes trace their ancestry back to a particular vode
named Rudy. Rudy is still quite alive, as is every one of his many descendants. Vodes
reproduce asexually, like bees; each vode has exactly one parent (except Rudy, who has no
parent). There are three dierent colors of vodescyan, magenta, and yellow. The color of
each vode is correlated exactly with the number and colors of its children, as follows:
Each cyan vode has two children, exactly one of which is yellow.
Each yellow vode has exactly one child, which is not yellow.
Magenta vodes have no children.
In each of the following problems, let C , M , and Y respectively denote the number of cyan,
magenta, and yellow vodes on Ygdrasil.
(a) Prove that M = C + 1.
(b) Prove that either Y = C or Y = M .
(c) Prove that Y = M if and only if Rudy is yellow.
[Hint: Be very careful to prove that you have considered all possibilities.]
3
CS 473G
Homework 0 (due September 1, 2005)
Fall 2005
6. [Extra credit]3
Lazy binary is a variant of standard binary notation for representing natural numbers where
we allow each bit to take on one of three values: 0, 1, or 2. Lazy binary notation is dened
inductively as follows.
The lazy binary representation of zero is 0.
Given the lazy binary representation of any non-negative integer n, we can construct the
lazy binary representation of n + 1 as follows:
(a) increment the rightmost digit;
(b) if any digit is equal to 2, replace the rightmost 2 with 0 and increment the digit
immediately to its left.
Here are the rst several natural numbers in lazy binary notation:
0, 1, 10, 11, 20, 101, 110, 111, 120, 201, 210, 1011, 1020, 1101, 1110, 1111, 1120, 1201,
1210, 2011, 2020, 2101, 2110, 10111, 10120, 10201, 10210, 11011, 11020, 11101, 11110,
11111, 11120, 11201, 11210, 12011, 12020, 12101, 12110, 20111, 20120, 20201, 20210, 21011,
21020, 21101, 21110, 101111, 101120, 101201, 101210, 102011, 102020, 102101, 102110, . . .
(a) Prove that in any lazy binary number, between any two 2s there is at least one 0, and
between two 0s there is at least one 2.
(b) Prove that for any natural number N , the sum of the digits of the lazy binary representation of N is exactly lg(N + 1) .
3
The I dont know rule does not apply to extra credit problems. There is no such thing as partial extra credit.
4
CS 473G
Homework 0 (due September 1, 2005)
Fall 2005
Practice Problems
The remaining problems are for practice only. Please do not submit solutions. On the other hand,
feel free to discuss these problems in oce hours or on the course newsgroup.
1. Sort the functions in each box from asymptotically smallest to asymptotically largest, indicating ties if there are any. You do not need to turn in proofs (in fact, please dont turn in
proofs), but you should do them anyway, just for practice.
1
2
lg n
lg n
lg n
2
2lg n
2
lg n
lg2 n
2lg n
lg n2
n
lg n
2
lg n2
n
lg 2 n
lg2 n
n2
n
lg 2
lg
2
n
2n
lg 2n
lg2 n
2
n
lg 2n
lg n
2
To simplify your answers, write f (n)
g (n) to mean f (n) = o(g (n)), and write f (n) g (n)
to mean f (n) = (g (n)). For example, the functions n2 , n, n , n3 could be sorted either as
2
n
n2
n3 .
n3 or as n
n
n2 n
2
2
2. Recall the standard recursive denition of the Fibonacci numbers: F0 = 0, F1 = 1, and
Fn = Fn1 + Fn2 for all n 2. Prove the following identities for all positive integers
n and m.
(a) Fn is even if and only if n is divisible by 3.
n
(b)
(c)
Fi = Fn+2 1
i=0
2
Fn Fn+1 Fn1 =
(1)n+1
(d) If n is an integer multiple of m, then Fn is an integer multiple of Fm .
3. Penn and Teller have a special deck of fty-two cards, with no face cards and nothing but
clubsthe ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, . . . , 52 of clubs. (Theyre big cards.) Penn
shues the deck until each each of the 52! possible orderings of the cards is equally likely. He
then takes cards one at a time from the top of the deck and gives them to Teller, stopping as
soon as he gives Teller the three of clubs.
(a) On average, how many cards does Penn give Teller?
(b) On average, what is the smallest-numbered card that Penn gives Teller?
(c) On average, what is the largest-numbered card that Penn gives Teller?
Prove that your answers are correct. (If you have to appeal to intuition or common sense,
your answers are probably wrong.) [Hint: Solve for an n-card deck, and then set n to 52.]
5
CS 473G
Homework 0 (due September 1, 2005)
Fall 2005
4. Algorithms and data structures were developed millions of years ago by the Martians, but
not quite in the same way as the recent development here on Earth. Intelligent life evolved
independently on Mars two moons, Phobos and Deimos.4 When the two races nally met on
the surface of Mars, after thousands of years of separate philosophical, cultural, religious, and
scientic development, their disagreements over the proper structure of binary search trees
led to a bloody (or more accurately, ichorous) war, ultimately leading to the destruction of
all Martian life.
A Phobian binary search tree is a full binary tree that stores a set X of search keys. The
root of the tree stores the smallest element in X . If X has more than one element, then the
left subtree stores all the elements less than some pivot value p, and the right subtree stores
everything else. Both subtrees are nonempty Phobian binary search trees. The actual pivot
value p is never stored in the tree.
A
A
I
A
A
C
B
C
C
H
E
S
I
I
I
N
M
N
S
R
T
T
Y
A Phobian binary search tree for the set {M, A, R, T, I, N, B, Y, S, E, C, H}.
(a) Describe and analyze an algorithm Find(x, T ) that returns True if x is stored in the
Phobian binary search tree T , and False otherwise.
(b) A Deimoid binary search tree is almost exactly the same as its Phobian counterpart,
except that the largest element is stored at the root, and both subtrees are Deimoid
binary search trees. Describe and analyze an algorithm to transform an n-node Phobian binary search tree into a Deimoid binary search tree in O(n) time, using as little
additional space as possible.
5. Tatami are rectangular mats used to tile oors in traditional Japanese houses. Exact dimensions of tatami mats vary from one region of Japan to the next, but they are always twice as
long in one dimension than in the other. (In Tokyo, the standard size is 180cm90cm.)
(a) How many dierent ways are there to tile a 2 n rectangular room with 1 2 tatami
mats? Set up a recurrence and derive an exact closed-form solution. [Hint: The answer
involves a familiar recursive sequence.]
(b) According to tradition, tatami mats are always arranged so that four corners never meet.
How many dierent traditional ways are there to tile a 3 n rectangular room with 1 2
tatami mats? Set up a recurrence and derive an exact closed-form solution.
(c) How many dierent traditional ways are there to tile an n n square with 1 2 tatami
mats? Prove your answer is correct.
4
Greek for fear and panic, respectively. Doesnt that make you feel better?
6
CS 473G: Combinatorial Algorithms, Fall 2005
Homework 1
Due Tuesday, September 13, 2005, by midnight (11:59:59pm CDT)
Name:
Net ID:
Alias:
Name:
Net ID:
Alias:
Name:
Net ID:
Alias:
Starting with Homework 1, homeworks may be done in teams of up to three people. Each team
turns in just one solution, and every member of a team gets the same grade.
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes
above. Staple this sheet to the top of your answer to problem 1.
There are two steps required to prove NP-completeness: (1) Prove that the problem is in NP,
by describing a polynomial-time verication algorithm. (2) Prove that the problem is NP-hard,
by describing a polynomial-time reduction from some other NP-hard problem. Showing that the
reduction is correct requires proving an if-and-only-if statement; dont forget to prove both the if
part and the only if part.
Required Problems
1. Some NP-Complete problems
(a) Show that the problem of deciding whether one graph is a subgraph of another is NPcomplete.
(b) Given a boolean circuit that embeds in the plane so that no 2 wires cross, PlanarCircuitSat is the problem of determining if there is a boolean assignment to the inputs
that makes the circuit output true. Prove that PlanarCircuitSat is NP-Complete.
(c) Given a set S with 3n numbers, 3partition is the problem of determining if S can be
partitioned into n disjoint subsets, each with 3 elements, so that every subset sums to the
same value. Given a set S and a collection of three element subsets of S , X3M (or exact
3-dimensional matching ) is the problem of determining whether there is a subcollection
of n disjoint triples that exactly cover S .
Describe a polynomial-time reduction from 3partition to X3M.
CS 473G
Homework 1 (due September 13, 2005)
Fall 2005
(d) A domino is a 1 2 rectangle divided into two squares, each of which is labeled with an
integer.1 In a legal arrangement of dominoes, the dominoes are lined up end-to-end so
that the numbers on adjacent ends match.
A legal arrangement of dominos, where every integer between 1 and 6 appears twice
Prove that the following problem is NP-complete: Given an arbitrary collection D of
dominoes, is there a legal arrangement of a subset of D in which every integer between
1 and n appears exactly twice?
2. Prove that the following problems are all polynomial-time equivalent, that is, if any of these
problems can be solved in polynomial time, then all of them can.
Clique: Given a graph G and an integer k , does there exist a clique of size k in G?
FindClique: Given a graph G and an integer k , nd a clique of size k in G if one exists.
MaxClique: Given a graph G, nd the size of the largest clique in the graph.
FindMaxClique: Given a graph G, nd a clique of maximum size in G.
3. Consider the following problem: Given a set of n points in the plane, nd a set of line segments
connecting the points which form a closed loop and do not intersect each other.
Describe a linear time reduction from the problem of sorting n numbers to the problem
described above.
4. In graph coloring, the vertices of a graph are assigned colors so that no adjacent vertices
recieve the same color. We saw in class that determining if a graph is 3-colorable is NPComplete.
Suppose you are handed a magic black box that, given a graph as input, tells you in constant
time whether or not the graph is 3-colorable. Using this black box, give a polynomial-time
algorithm to 3-color a graph.
5. Suppose that Cook had proved that graph coloring was NP-complete rst, instead of CircuitSAT. Using only the fact that graph coloring is NP-complete, show that CircuitSAT
is NP-complete.
1
These integers are usually represented by pips, exactly like dice. On a standard domino, the number of pips on
each side is between 0 and 6; we will allow arbitrary integer labels. A standard set of dominoes has one domino for
each possible unordered pair of labels; we do not require that every possible label pair is in our set.
2
CS 473G
Homework 1 (due September 13, 2005)
Fall 2005
Practice Problems
1. Given an initial conguration consisting of an undirected graph G = (V, E ) and a function
p : V IN indicating an initial number of pebbles on each vertex, Pebble-Destruction
asks if there is a sequence of pebbling moves starting with the initial conguration and ending
with a single pebble on only one vertex of V . Here, a pebbling move consists of removing
two pebbles from a vertex v and adding one pebble to a neighbor of v . Prove that PebbleDestruction is NP-complete.
2. Consider nding the median of 5 numbers by using only comparisons. What is the exact worst
case number of comparisons needed to nd the median? To prove your answer is correct, you
must exhibit both an algorithm that uses that many comparisons and a proof that there is
no faster algorithm. Do the same for 6 numbers.
3. Partition is the problem of deciding, given a set S of numbers, whether it can be partitioned
into two subsets whose sums are equal. (A partition of S is a collection of disjoint subsets
whose union is S .) SubsetSum is the problem of deciding, given a set S of numbers and a
target sum t, whether any subset of number in S sum to t.
(a) Describe a polynomial-time reduction from SubsetSum to Partition.
(b) Describe a polynomial-time reduction from Partition to SubsetSum.
4. Recall from class that the problem of deciding whether a graph can be colored with three
colors, so that no edge joins nodes of the same color, is NP-complete.
(a) Using the gadget in Figure 1(a), prove that deciding whether a planar graph can be 3colored is NP-complete. [Hint: Show that the gadget can be 3-colored, and then replace
any crossings in a planar embedding with the gadget appropriately.]
(a)
(b)
Figure 1. (a) Gadget for planar 3-colorability. (b) Gadget for degree-4 planar 3-colorability.
(b) Using the previous result and the gadget in gure 1(b), prove that deciding whether a
planar graph with maximum degree 4 can be 3-colored is NP-complete. [Hint: Show
that you can replace any vertex with degree greater than 4 with a collection of gadgets
connected in such a way that no degree is greater than four.]
3
CS 473G
Homework 1 (due September 13, 2005)
Fall 2005
5. (a) Prove that if G is an undirected bipartite graph with an odd number of vertices, then
G is nonhamiltonian. Describe a polynomial-time algorithm to nd a hamiltonian cycle
in an undirected bipartite graph, or establish that no such cycle exists.
(b) Describe a polynomial time algorithm to nd a hamiltonian path in a directed acyclic
graph, or establish that no such path exists.
(c) Why dont these results imply that P=NP?
6. Consider the following pairs of problems:
(a) MIN SPANNING TREE and MAX SPANNING TREE
(b) SHORTEST PATH and LONGEST PATH
(c) TRAVELING SALESMAN PROBLEM and VACATION TOUR PROBLEM (the longest
tour is sought).
(d) MIN CUT and MAX CUT (between s and t)
(e) EDGE COVER and VERTEX COVER
(f) TRANSITIVE REDUCTION and MIN EQUIVALENT DIGRAPH
(all of these seem dual or opposites, except the last, which are just two versions of minimal
representation of a graph).
Which of these pairs are polytime equivalent and which are not? Why?
7. Prove that Primality (Given n, is n prime?) is in NP co-NP. [Hint: co-NP is easyWhats
a certicate for showing that a number is composite? For NP, consider a certicate involving
primitive roots and recursively their primitive roots. Show that this tree of primitive roots
can be veried an used to show that n is prime in polynomial time.]
8. How much wood would a woodchuck chuck if a woodchuck could chuck wood?
4
CS 473G: Combinatorial Algorithms, Fall 2005
Homework 2
Due Thursday, September 22, 2005, by midnight (11:59:59pm CDT)
Name:
Net ID:
Alias:
Name:
Net ID:
Alias:
Name:
Net ID:
Alias:
Starting with Homework 1, homeworks may be done in teams of up to three people. Each team
turns in just one solution, and every member of a team gets the same grade.
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above.
Staple this sheet to the top of your homework.
Required Problems
1. (a) Suppose Lois has an algorithm to compute the shortest common supersequence of two
arrays of integers in O(n) time. Describe an O(n log n)-time algorithm to compute
the longest common subsequence of two arrays of integers, using Loiss algorithm as a
subroutine.
(b) Describe an O(n log n)-time algorithm to compute the longest increasing subsequence of
an array of integers, using Loiss algorithm as a subroutine.
(c) Now suppose Lisa has an algorithm that can compute the longest increasing subsequence
of an array of integers in O(n) time. Describe an O(n log n)-time algorithm to compute
the longest common subsequence of two arrays A[1 .. n] and B [1 .. n] of integers, where
A[i] = A[j ] for all i = j , using Lisas algorithm as a subroutine.1
1
For extra credit, remove the assumption that the elements of A are distinct. This is probably impossible.
CS 473G
Homework 2 (due September 22, 2005)
Fall 2005
2. In a previous incarnation, you worked as a cashier in the lost 19th-century Antarctican colony
of Nadira, spending the better part of your day giving change to your customers. Because
paper is a very rare and valuable resource on Antarctica, cashiers were required by law to use
the fewest bills possible whenever they gave change. Thanks to the numerological predilections
of one of its founders, the currency of Nadira, called Dream Dollars, was available in the
following denominations: $1, $4, $7, $13, $28, $52, $91, $365.2
(a) The greedy change algorithm repeatedly takes the largest bill that does not exceed the
target amount. For example, to make $122 using the greedy algorithm, we rst take a
$91 bill, then a $28 bill, and nally three $1 bills. Give an example where this greedy
algorithm uses more Dream Dollar bills than the minimum possible.
(b) Describe and analyze an ecient algorithm that computes, given an integer n, the minimum number of bills needed to make n Dream Dollars.
3. Scientists have branched out from the bizarre planet of Yggdrasil to study the vodes which
have settled on Ygdrasils moon, Xryltcon. All vodes on Xryltcon are descended from the rst
vode to arrive there, named George. Each vode has a color, either cyan, magenta, or yellow,
but breeding patterns are not the same as on Yggdrasil; every vode, regardless of color, has
either two children (with arbitrary colors) or no children.
George and all his descendants are alive and well, and they are quite excited to meet
the scientists who wish to study them. Unsurprisingly, these vodes have had some strange
mutations in their isolation on Xryltcon. Each vode has a weirdness rating; weirder vodes are
more interesting to the visiting scientists. (Some vodes even have negative weirdness ratings;
they make other vodes more boring just by standing next to them.)
Also, Xryltconian society is strictly governed by a number of sacred cultural traditions.
No cyan vode may be in the same room as its non-cyan children (if it has any).
No magenta vode may be in the same room as its parent (if it has one).
Each yellow vode must be attended at all times by its grandchildren (if it has any).
George must be present at any gathering of more than fty vodes.
The scientists have exactly one chance to study a group of vodes in a single room. You
are given the family tree of all the vodes on Xryltcon, along with the wierdness value of each
vode. Design and analyze an ecient algorithm to decide which vodes the scientists should
invite to maximize the sum of the wierdness values of the vodes in the room. Be careful to
respect all of the vodes cultural taboos.
2
For more details on the history and culture of Nadira, including images of the various denominations of Dream
Dollars, see http://www.dream-dollars.com. Really.
2
CS 473G
Homework 2 (due September 22, 2005)
Fall 2005
4. A subtree of a (rooted, ordered) binary tree T consists of a node and all its descendants.
Design and analyze an ecient algorithm to compute the largest common subtree of two
given binary trees T1 and T2 , that is, the largest subtree of T1 that is isomorphic to a subtree
in T2 . The contents of the nodes are irrelevant; we are only interested in matching the
underlying combinatorial structure.
Two binary trees, with their largest common subtree emphasized
5. Let D[1 .. n] be an array of digits, each an integer between 0 and 9. An digital subsequence
of D is an sequence of positive integers composed in the usual way from disjoint substrings
of D. For example, 3, 4, 5, 6, 23, 38, 62, 64, 83, 279 is an increasing digital subsequence of the
rst several digits of :
3 , 1, 4 , 1, 5 , 9, 6 , 2, 3 , 4, 3, 8 , 4, 6, 2 , 6, 4 , 3, 3, 8, 3 , 2, 7, 9
The length of a digital subsequence is the number of integers it contains, not the number of
digits; the previous example has length 10.
Describe and analyze an ecient algorithm to compute the longest increasing digital
subsequence of D. [Hint: Be careful about your computational assumptions. How long does
it take to compare two k -digit numbers?]
6. [Extra credit] The chromatic number of a graph G is the minimum number of colors needed
to color the nodes of G so that no pair of adjacent nodes have the same color.
(a) Describe and analyze a recursive algorithm to compute the chromatic number of an
n-vertex graph in O(4n poly(n)) time. [Hint: Catalan numbers play a role here.]
(b) Describe and analyze an algorithm to compute the chromatic number of an n-vertex
graph in O(3n poly(n)) time. [Hint: Use dynamic programming. What is (1 + x)n ?]
(c) Describe and analyze an algorithm to compute the chromatic number of an n-vertex
graph in O((1 + 31/3 )n poly(n)) time. [Hint: Use (but dont regurgitate) the algorithm
in the lecture notes that counts all the maximal independent sets in an n-vertex graph
in O(3n/3 ) time.]
3
CS 473G
Homework 2 (due September 22, 2005)
Fall 2005
Practice Problems
1. Describe an algorithm to solve 3SAT in time O(n poly(n)), where = (1 + 5)/2. [Hint:
Prove that in each recursive call, either you have just eliminated a pure literal, or the formula
has a clause with at most two literals.]
2. Describe and analyze an algorithm to compute the longest increasing subsequence in an
n-element array of integers in O(n log n) time. [Hint: Modify the O(n2 )-time algorithm
presented in class.]
3. The edit distance between two strings A and B , denoted Edit(A, B ), is the minimum number
of insertions, deletions, or substitutions required to transform A into B (or vice versa). Edit
distance is sometimes also called the Levenshtein distance.
Let A = {A1 , A2 , . . . , Ak } be a set of strings. The edit radius of A is the minimum over
all strings X of the maximum edit distance from X to any string Ai :
EditRadius(A) =
min
max Edit(X, Ai )
strings X 1ik
A string X that achieves this minimum is called an edit center of A. A set of strings may
have several edit centers, but the edit radius is unique.
Describe an ecient algorithm to compute the edit radius of three given strings.
4. Given 5 sequences of numbers, each of length n, design and analyze an ecent algorithm to
compute the longest common subsequence among all 5 sequences.
5. Suppose we want to display a paragraph of text on a computer screen. The text consists of n
words, where the ith word is W [i] pixels wide. We want to break the paragraph into several
lines, each exactly L pixels long. Depending on which words we put on each line, we will
need to insert dierent amounts of white space between the words. The paragraph should
be fully justied, meaning that the rst word on each line starts at its leftmost pixel, and
except for the last line, the last character on each line ends at its rightmost pixel. (Look at
the paragraph you are reading right now!) There must be at least one pixel of white space
between any two words on the same line. Thus, if a line contains words i through j , then the
amount of extra white space on that line is L j + i j =i W [k ].
k
Dene the slop of a paragraph layout as the sum, over all lines except the last, of the cube
of the extra white space in each line. Describe an ecient algorithm to layout the paragraph
with minimum slop, given the list W [1 .. n] of word widths as input. You can assume that
W [i] < L/2 for each i, so that each line contains at least two words.
4
CS 473G
Homework 2 (due September 22, 2005)
Fall 2005
6. A partition of a positive integer n is a multiset of positive integers that sum to n. Traditionally,
the elements of a partition are written in non-decreasing order, separated by + signs. For
example, the integer 7 has exactly twelve partitions:
1+1+1+1+1+1+1 3+1+1+1+1 4+1+1+1
2+1+1+1+1+1
3+2+1+1
4+2+1
2+2+1+1+1
3+2+2
4+3
2+2+2+1
3+3+1
7
The roughness of a partition a1 + a2 + + ak is dened as follows:
k 1
(a1 + a2 + + ak ) =
|ai+1 ai 1| + ak 1
i=1
A smoothest partition of n is the partition of n with minimum roughness. Intuitively, the
smoothest partition is the one closest to a descending arithmetic series k + + 3 + 2 + 1,
which is the only partition that has roughness 0. For example, the smoothest partitions of 7
are 4 + 2 + 1 and 3 + 2 + 1 + 1:
(1 + 1 + 1 + 1 + 1 + 1 + 1) = 6 (3 + 1 + 1 + 1 + 1) = 4 (4 + 1 + 1 + 1) = 4
(2 + 1 + 1 + 1 + 1 + 1) = 4
(3 + 2 + 1 + 1) = 1
(4 + 2 + 1) = 1
(2 + 2 + 1 + 1 + 1) = 3
(3 + 2 + 2) = 2
(4 + 3) = 2
(2 + 2 + 2 + 1) = 2
(3 + 3 + 1) = 2
(7) = 7
Describe and analyze an algorithm to compute, given a positive integer n, a smoothest
partition of n.
5
CS 473G: Combinatorial Algorithms, Fall 2005
Homework 3
Due Tuesday, October 18, 2005, at midnight
Name:
Net ID:
Name:
Net ID:
Name:
Net ID:
Alias:
Alias:
Alias:
Starting with Homework 1, homeworks may be done in teams of up to three people. Each team
turns in just one solution, and every member of a team gets the same grade.
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above.
Staple this sheet to the top of your homework.
1. Consider the following greedy approximation algorithm to nd a vertex cover in a graph:
GreedyVertexCover(G):
C
while G has at least one edge
v vertex in G with maximum degree
GG\v
C C v
return C
In class we proved that the approximation ratio of this algorithm is O(log n); your task is to
prove a matching lower bound. Specically, prove that for any integer n, there is a graph G
with n vertices such that GreedyVertexCover(G) returns a vertex cover that is (log n)
times larger than optimal.
2. Prove that for any constant k and graph any coloring algorithm A, there is a graph G such
that A(G) > OP T (G) + k, where A(G) is the number of colors generated by algorithm A
for graph G, and OP T (G) is the optimal number of colors for G.
[Note: This does not contradict the possibility of a constant factor approximation algorithm.]
CS 473G
Homework 3 (due October 18, 2005)
Fall 2005
3. Let R be a set of rectangles in the plane, with horizontal and vertical edges. A stabbing set
for R is a set of points S such that every rectangle in R contains at least one point in S . The
rectangle stabbing problem asks, given a set R of rectangles, for the smallest stabbing set S .
(a) Prove that the rectangle stabbing problem is NP-hard.
(b) Describe and analyze an ecient approximation algorithm for the rectangle stabbing
problem. Give bounds on the approximation ratio of your algorithm.
4. Consider the following approximation scheme for coloring a graph G.
TreeColor(G):
T any spanning tree of G
Color the tree T with two colors
c2
for each edge (u, v ) G \ T
T T {(u, v )}
if color(u) = color(v )
Try recoloring u with an existing color
for i 1 to c
if no neighbor of u in T has color i
color(u) i
if color(u) = color(v )
Try recoloring v with an existing color
for i 1 to c
if no neighbor of v in T has color i
color(v ) i
if color(u) = color(v )
cc+1
color(u) c
Give up and use a new color
return c
(a) Prove that this algorithm correctly colors any bipartite graph.
(b) Prove an upper bound C on the number of colors used by this algorithm. Give a sample
graph and run that requires C colors.
(c) Does this algorithm approximate the minimum number of colors up to a constant factor?
In other words, is there a constant such that TreeColor(G) < OP T (G) for any
graph G? Justify your answer.
2
CS 473G
Homework 3 (due October 18, 2005)
Fall 2005
5. In the bin packing problem, we are given a set of n items, each with weight between 0 and 1,
and we are asked to load the items into as few bins as possible, such that the total weight in
each bin is at most 1. Its not hard to show that this problem is NP-Hard; this question asks
you to analyze a few common approximation algorithms. In each case, the input is an array
W [1 .. n] of weights, and the output is the number of bins used.
FirstFit(W [1 .. n]):
b0
NextFit(W [1 .. n]):
b0
T otal[0]
for i 1 to n
j 1; f ound False
while j b and f ound = False
if T otal[j ] + W [i] 1
T otal[j ] T otal[j ] + W [i]
f ound True
j j+1
for i 1 to n
if T otal[b] + W [i] > 1
bb+1
T otal[b] W [i]
else
T otal[b] T otal[b] + W [i]
if f ound = False
bb+1
T otal[b] = W [i]
return b
return b
(a) Prove that NextFit uses at most twice the optimal number of bins.
(b) Prove that FirstFit uses at most twice the optimal number of bins.
(c) Prove that if the weight array W is initially sorted in decreasing order, then FirstFit
uses at most (4 OP T + 1)/3 bins, where OP T is the optimal number of bins. The
following facts may be useful (but you need to prove them if your proof uses them):
In the packing computed by FirstFit, every item with weight more than 1/3 is
placed in one of the rst OP T bins.
FirstFit places at most OP T 1 items outside the rst OP T bins.
3
CS 473G: Combinatorial Algorithms, Fall 2005
Homework 4
Due Thursday, October 27, 2005, at midnight
Name:
Net ID:
Name:
Net ID:
Name:
Net ID:
Name:
Net ID:
Alias:
Alias:
Alias:
Alias:
Homeworks may be done in teams of up to three people. Each team turns in just one solution;
every member of a team gets the same grade.
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above.
Staple this sheet to the top of your solution to problem 1.
If you are an I2CS student, print (I2CS) next to your name. Teams that include
both on-campus and I2CS students can have up to four members. Any team containing
both on-campus and I2CS students automatically receives 3 points of extra credit.
For the rest of the semester, unless specically stated otherwise, you
may assume that the function Random(m) returns an integer chosen
uniformly at random from the set {1, 2, , . . . , m} in O(1) time. For
example, a fair coin ip is obtained by calling Random(2).
1. Consider the following randomized algorithm for choosing the largest bolt. Draw a bolt
uniformly at random from the set of n bolts, and draw a nut uniformly at random from the
set of n nuts. If the bolt is smaller than the nut, discard the bolt, draw a new bolt uniformly
at random from the unchosen bolts, and repeat. Otherwise, discard the nut, draw a new nut
uniformly at random from the unchosen nuts, and repeat. Stop either when every nut has
been discarded, or every bolt except the one in your hand has been discarded.
What is the exact expected number of nut-bolt tests performed by this algorithm? Prove
your answer is correct. [Hint: What is the expected number of unchosen nuts and bolts when
the algorithm terminates?]
CS 473G
Homework 4 (due October 26, 2005)
Fall 2005
2. A meldable priority queue stores a set of keys from some totally-ordered universe (such as the
integers) and supports the following operations:
MakeQueue: Return a new priority queue containing the empty set.
FindMin(Q): Return the smallest element of Q (if any).
DeleteMin(Q): Remove the smallest element in Q (if any).
Insert(Q, x): Insert element x into Q, if it is not already there.
DecreaseKey(Q, x, y ): Replace an element x Q with a smaller key y . (If y > x, the
operation fails.) The input is a pointer directly to the node in Q containing x.
Delete(Q, x): Delete the element x Q. The input is a pointer directly to the node
in Q containing x.
Meld(Q1 , Q2 ): Return a new priority queue containing all the elements of Q1 and Q2 ;
this operation destroys Q1 and Q2 .
A simple way to implement such a data structure is to use a heap-ordered binary tree, where
each node stores a key, along with pointers to its parent and two children. Meld can be
implemented using the following randomized algorithm:
Meld(Q1 , Q2 ):
if Q1 is empty return Q2
if Q2 is empty return Q1
if key (Q1 ) > key (Q2 )
swap Q1 Q2
with probability 1/2
left (Q1 ) Meld(left (Q1 ), Q2 )
else
right (Q1 ) Meld(right (Q1 ), Q2 )
return Q1
(a) Prove that for any heap-ordered binary trees Q1 and Q2 (not just those constructed by
the operations listed above), the expected running time of Meld(Q1 , Q2 ) is O(log n),
where n = |Q1 | + |Q2 |. [Hint: How long is a random root-to-leaf path in an n-node
binary tree if each left/right choice is made with equal probability?]
(b) Prove that Meld(Q1 , Q2 ) runs in O(log n) time with high probability.
(c) Show that each of the other meldable priority queue operations cab be implemented with
at most one call to Meld and O(1) additional time. (This implies that every operation
takes O(log n) time with high probability.)
2
CS 473G
Homework 4 (due October 26, 2005)
Fall 2005
3. Let M [1 .. n][1 .. n] be an n n matrix in which every row and every column is sorted. Such
an array is called totally monotone. No two elements of M are equal.
(a) Describe and analyze an algorithm to solve the following problem in O(n) time: Given
indices i, j, i , j as input, compute the number of elements of M smaller than M [i][j ]
and larger than M [i ][j ].
(b) Describe and analyze an algorithm to solve the following problem in O(n) time: Given
indices i, j, i , j as input, return an element of M chosen uniformly at random from the
elements smaller than M [i][j ] and larger than M [i ][j ]. Assume the requested range is
always non-empty.
(c) Describe and analyze a randomized algorithm to compute the median element of M in
O(n log n) expected time.
4. Let X [1 .. n] be an array of n distinct real numbers, and let N [1 .. n] be an array of indices
with the following property: If X [i] is the largest element of X , then X [N [i]] is the smallest
element of X ; otherwise, X [N [i]] is the smallest element of X that is larger than X [i].
For example:
i123456789
X [i] 83 54 16 31 45 99 78 62 27
N [i] 6 8 9 5 2 3 1 7 4
Describe and analyze a randomized algorithm that determines whether a given number x
appears in the array X in O( n) expected time. Your algorithm may not modify the
arrays X and N ext.
5. A majority tree is a complete ternary tree with depth n, where every leaf is labeled either 0
or 1. The value of a leaf is its label; the value of any internal node is the majority of the
values of its three children. Consider the problem of computing the value of the root of a
majority tree, given the sequence of 3n leaf labels as input. For example, if n = 2 and the
leaves are labeled 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, the root has value 0.
A majority tree with depth n = 2.
(a) Prove that any deterministic algorithm that computes the value of the root of a majority
tree must examine every leaf. [Hint: Consider the special case n = 1. Recurse.]
(b) Describe and analyze a randomized algorithm that computes the value of the root in
worst-case expected time O(cn ) for some constant c < 3. [Hint: Consider the special
case n = 1. Recurse.]
3
CS 473G
Homework 4 (due October 26, 2005)
Fall 2005
6. [Extra credit] In the usual theoretical presentation of treaps, the priorities are random real
numbers chosen uniformly from the interval [0, 1], but in practice, computers only have access
to random bits. This problem asks you to analyze a modication of treaps that takes this
limitation into account.
Suppose the priority of a node v is abstractly represented as an innite sequence v [1 .. ] of
random bits, which is interpreted as the rational number
v [i] 2i .
priority (v ) =
i=1
However, only a nite number v of these bits are actually known at any given time. When a
node v is rst created, none of the priority bits are known: v = 0. We generate (or reveal)
new random bits only when they are necessary to compare priorities. The following algorithm
compares the priorities of any two nodes in O(1) expected time:
LargerPriority(v, w):
for i 1 to
if i > v
v i; v [i] RandomBit
if i >
w
w
i; w [i] RandomBit
if v [i] > w [i]
return v
else if v [i] < w [i]
return w
Suppose we insert n items one at a time into an initially empty treap. Let L =
vv
denote the total number of random bits generated by calls to LargerPriority during these
insertions.
(a) Prove that E [L] = (n).
(b) Prove that E [ v ] = (1) for any node v . [Hint: This is equivalent to part (a). Why?]
(c) Prove that E [
root ]
= (log n). [Hint: Why doesnt this contradict part (b)?]
4
CS 473G: Combinatorial Algorithms, Fall 2005
Homework 5
Due Thursday, November 17, 2005, at midnight
(because you really dont want homework due over Thanksgiving break)
Name:
Net ID:
Name:
Net ID:
Name:
Net ID:
Name:
Net ID:
Alias:
Alias:
Alias:
Alias:
Homeworks may be done in teams of up to three people. Each team turns in just one solution;
every member of a team gets the same grade.
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above.
Attach this sheet (or the equivalent information) to the top of your solution to problem 1.
If you are an I2CS student, print (I2CS) next to your name. Teams that include
both on-campus and I2CS students can have up to four members. Any team containing
both on-campus and I2CS students automatically receives 3 points of extra credit.
Problems labeled are likely to require techniques from next weeks lectures on cuts, ows, and
matchings. See also Chapter 7 in Kleinberg and Tardos, or Chapter 26 in CLRS.
1. Suppose you are asked to construct the minimum spanning tree of a graph G, but you are not
completely sure of the edge weights. Specically, you have a conjectured weight w(e) for every
edge e in the graph, but you also know that up to k of these conjectured weights are wrong.
With the exception of one edge e whose true weight you know exactly, you dont know which
edges are wrong, or even how theyre wrong; the true weights of those edges could be larger
or smaller than the conjectured weights. Given this unreliable information, it is of course
impossible to reliably construct the true minimum spanning tree of G, but it is still possible
to say something about your special edge.
Describe and analyze an ecient algorithm to determine whether a specic edge e, whose
actual weight is known, is denitely not in the minimum spanning tree of G under the stated
conditions. The input consists of the graph G, the conjectured weight function w : E (G) IR,
the positive integer k , and the edge e.
CS 473G
Homework 5 (due November 17, 2005)
Fall 2005
2. Most classical minimum-spanning-tree algorithms use the notions of safe and useless edges
described in the lecture notes, but there is an alternate formulation. Let G be a weighted
undirected graph, where the edge weights are distinct. We say that an edge e is dangerous if
it is the longest edge in some cycle in G, and useful if it does not lie in any cycle in G.
(a) Prove that the minimum spanning tree of G contains every useful edge.
(b) Prove that the minimum spanning tree of G does not contain any dangerous edge.
(c) Describe and analyze an ecient implementation of the anti-Kruskal MST algorithm:
Examine the edges of G in decreasing order; if an edge is dangerous, remove it from G.
[Hint: It wont be as fast as the algorithms you saw in class.]
3. The UIUC Computer Science department has decided to build a mini-golf course in the
basement of the Siebel Center! The playing eld is a closed polygon bounded by m horizontal
and vertical line segments, meeting at right angles. The course has n starting points and n
holes, in one-to-one correspondence. It is always possible hit the ball along a straight line
directly from each starting point to the corresponding hole, without touching the boundary of
the playing eld. (Players are not allowed to bounce golf balls o the walls; too much glass.)
The n starting points and n holes are all at distinct locations.
Sadly, the architects computer crashed just as construction was about to begin. Thanks to
the herculean eorts of their sysadmins, they were able to recover the locations of the starting
points and the holes, but all information about which starting points correspond to which
holes was lost!
Describe and analyze an algorithm to compute a one-to-one correspondence between the
starting points and the holes that meets the straight-line requirement, or to report that no
such correspondence exists. The input consists of the x- and y -coordinates of the m corners
of the playing eld, the n starting points, and the n holes. Assume you can determine in
constant time whether two line segments intersect, given the x- and y -coordinates of their
endpoints.
A minigolf course with ve starting points ( ) and ve holes (), and a legal correspondence between them.
4. Let G = (V, E ) be a directed graph where the in-degree of each vertex is equal to its outdegree. Prove or disprove the following claim: For any two vertices u and v in G, the number
of mutually edge-disjoint paths from u to v is equal to the number of mutually edge-disjoint
paths from v to u.
2
CS 473G
Homework 5 (due November 17, 2005)
Fall 2005
5. You are given a set of n boxes, each specied by its height, width, and depth. The order of
the dimensions is unimportant; for example, a 1 2 3 box is exactly the same as a 3 1 2
box of a 2 1 3 box. You can nest box A inside box B if and only if A can be rotated so
that it has strictly smaller height, strictly smaller width, and strictly smaller depth than B .
(a) Design and analyze an ecient algorithm to determine the largest sequence of boxes that
can be nested inside one another. [Hint: Model the nesting relationship as a graph.]
(b) Describe and analyze an ecient algorithm to nest all n boxes into as few groups as
possible, where each group consists of a nested sequence. You are not allowed to put
two boxes side-by-side inside a third box, even if they are small enough to t.1 [Hint:
Model the nesting relationship as a dierent graph.]
6. [Extra credit] Prove that Fords generic shortest-path algorithm (described in the lecture
notes) can take exponential time in the worst case when implemented with a stack instead
of a heap (like Dijkstra) or a queue (like Bellman-Ford). Specically, construct for every
positive integer n a weighted directed n-vertex graph Gn , such that the stack-based shortestpath algorithm call Relax (2n ) times when Gn is the input graph. [Hint: Towers of Hanoi.]
1
Without this restriction, the problem is NP-hard, even for one-dimensional boxes.
3
CS 473G: Combinatorial Algorithms, Fall 2005
Homework 6
Practice only; nothing to turn in.
1. A small airline, Ivy Air, ies between three cities: Ithaca (a small town in upstate New York),
Newark (an eyesore in beautiful New Jersey), and Boston (a yuppie town in Massachusetts).
They oer several ights but, for this problem, let us focus on the Friday afternoon ight
that departs from Ithaca, stops in Newark, and continues to Boston. There are three types
of passengers:
(a) Those traveling from Ithaca to Newark (god only knows why).
(b) Those traveling from Newark to Boston (a very good idea).
(c) Those traveling from Ithaca to Boston (it depends on who you know).
The aircraft is a small commuter plane that seats 30 passengers. The airline oers three fare
classes:
(a) Y class: full coach.
(b) B class: nonrefundable.
(c) M class: nonrefundable, 3-week advanced purchase.
Ticket prices, which are largely determined by external inuences (i.e., competitors), have
been set and advertised as follows:
Y
B
M
Ithaca-Newark
300
220
100
Newark-Boston
160
130
80
Ithaca-Boston
360
280
140
Based on past experience, demand forecasters at Ivy Air have determined the following upper
bounds on the number of potential customers in each of the 9 possible origin-destination/fareclass combinations:
Y
B
M
Ithaca-Newark
4
8
22
Newark-Boston
8
13
20
Ithaca-Boston
3
10
18
The goal is to decide how many tickets from each of the 9 origin/destination/fare-class combinations to sell. The constraints are that the place cannot be overbooked on either the two
legs of the ight and that the number of tickets made available cannot exceed the forecasted
maximum demand. The objective is to maximize the revenue.
Formulate this problem as a linear programming problem.
CS 473G
Homework 6
Fall 2005
2. (a) Suppose we are given a directed graph G = (V, E ), a length function : E IR, and
a source vertex s V . Write a linear program to compute the shortest-path distance
from s to every other vertex in V . [Hint: Dene a variable for each vertex representing
its distance from s. What objective function should you use?]
(b) In the minimum-cost multicommodity-ow problem, we are given a directed graph G =
(V, E ), in which each edge u v has an associated nonnegative capacity c(u v ) 0
and an associated cost (u v ). We are given k dierent commodities, each specied
by a triple Ki = (si , ti , di ), where si is the source node of the commodity, ti is the target
node for the commodity i, and di is the demand : the desired ow of commodity i from
si to ti . A ow for commodity i is a non-negative function fi : E IR0 such that
the total ow into any vertex other than si or ti is equal to the total ow out of that
vertex. The aggregate ow F : E IR is dened as the sum of these individual ows:
k
F (u v ) =
i=1 fi (u v ). The aggregate ow F (u v ) on any edge must not
exceed the capacity c(u v ). The goal is to nd an aggregate ow whose total cost
uv F (u v ) (u v ) is as small as possible. (Costs may be negative!) Express
this problem as a linear program.
3. In class we described the duality transformation only for linear programs in canonical form:
Primal ()
max c x
s.t. Ax b
x 0
Dual ( )
min y b
s.t. yA c
y 0
Describe precisely how to dualize the following more general linear programming problem:
d
maximize
cj xj
j =1
d
aij xj bi
for each i = 1 .. p
aij xj = bi
for each i = p + 1 .. p + q
aij xj bi
subject to
for each i = p + q + 1 .. n
j =1
d
j =1
d
j =1
Your dual problem should have one variable for each primal constraint, and the dual of your
dual program should be precisely the original linear program.
4. (a) Model the maximum-cardinality bipartite matching problem as a linear programming
problem. The input is a bipartite graph G = (U, V ; E ), where E U V ; the output is
the largest matching in G. Your linear program should have one variable for every edge.
(b) Now dualize the linear program from part (a). What do the dual variables represent?
What does the objective function represent? What problem is this!?
2
CS 473G
Homework 6
Fall 2005
5. An integer program is a linear program with the additional constraint that the variables must
take only integer values.
(a) Prove that deciding whether an integer program has a feasible solution is NP-complete.
(b) Prove that nding the optimal feasible solution to an integer program is NP-hard.
[Hint: Almost any NP=hard decision problem can be rephrased as an integer program. Pick
your favorite.]
6. Consider the LP formulation of the shortest path problem presented in class:
maximize
dt
subject to
ds = 0
dv du
uv
for every edge u v
Characterize the feasible bases for this linear program in terms of the original weighted
graph. What does a simplex pivoting operation represent? What is a locally optimal (i.e.,
dual feasible) basis? What does a dual pivoting operation represent?
7. Consider the LP formulation of the maximum-ow problem presented in class:
fsw
maximize
w
fus
u
fvw
subject to
w
fuv = 0
for every vertex v = s, t
u
fuv cuv for every edge u v
fuv 0
for every edge u v
Is the Ford-Fulkerson augmenting path algorithm an instance of the simplex algorithm applied
to this linear program? Why or why not?
8. Hellys theorem says that for any collection of convex bodies in IRn , if every n + 1 of them
intersect, then there is a point lying in the intersection of all of them. Prove Hellys theorem
for the special case that the convex bodies are halfspaces. [Hint: Show that if a system of
inequalities Ax b does not have a solution, then we can select n + 1 of the inequalities such
that the resulting system does not have a solution. Construct a primal LP from the system
by choosing a 0 cost vector.]
3
CS 473G
Midterm 1 Questions (September 27, 2005)
Fall 2005
You have 90 minutes to answer four of these questions.
Write your answers in the separate answer booklet.
You may take the question sheet with you when you leave.
1. You and your eight-year-old nephew Elmo decide to play a simple card game. At the beginning
of the game, the cards are dealt face up in a long row. Each card is worth a dierent number
of points. After all the cards are dealt, you and Elmo take turns removing either the leftmost
or rightmost card from the row, until all the cards are gone. At each turn, you can decide
which of the two cards to take. The winner of the game is the player that has collected the
most points when the game ends.
Having never taken an algorithms class, Elmo follows the obvious greedy strategywhen its
his turn, Elmo always takes the card with the higher point value. Your task is to nd a
strategy that will beat Elmo whenever possible. (It might seem mean to beat up on a little
kid like this, but Elmo absolutely hates it when grown-ups let him win.)
(a) Prove that you should not also use the greedy strategy. That is, show that there is a
game that you can win, but only if you do not follow the same greedy strategy as Elmo.
(b) Describe and analyze an algorithm to determine, given the initial sequence of cards, the
maximum number of points that you can collect playing against Elmo.
2. Suppose you are given a magical black box that can tell you in constant time whether or not
a given graph has a Hamiltonian cycle. Using this magic black box as a subroutine, describe
and analyze a polynomial-time algorithm to actually compute a Hamiltonian cycle in a given
graph, if one exists.
3. Let X be a set of n intervals on the real line. A subset of intervals Y X is called a tiling
path if the intervals in Y cover the intervals in X , that is, any real value that is contained in
some interval in X is also contained in some interval in Y . The size of a tiling cover is just
the number of intervals.
Describe and analyze an algorithm to compute the smallest tiling path of X as quickly as
possible. Assume that your input consists of two arrays XL [1 .. n] and XR [1 .. n], representing
the left and right endpoints of the intervals in X .
A set of intervals. The seven shaded intervals form a tiling path.
1
CS 473G
Midterm 1 Questions (September 27, 2005)
Fall 2005
4. Prove that the following problem is NP-complete: Given an undirected graph, does it have a
spanning tree in which every node has degree at most 3?
A graph with a spanning tree of maximum degree 3.
5. The Tower of Hanoi puzzle, invented by Edouard Lucas in 1883, consists of three pegs and
n disks of dierent sizes. Initially, all n disks are on the same peg, stacked in order by size,
with the largest disk on the bottom and the smallest disk on top. In a single move, you can
move the topmost disk on any peg to another peg; however, you are never allowed to place a
larger disk on top of a smaller one. Your goal is to move all n disks to a dierent peg.
(a) Prove that the Tower of Hanoi puzzle can be solved in exactly 2n 1 moves. [Hint:
Youve probably seen this before.]
(b) Now suppose the pegs are arranged in a circle and you are only allowed to move disks
counterclockwise. How many moves do you need to solve this restricted version of the
puzzle? Give a upper bound in the form O(f (n)) for some function f (n). Prove your
upper bound is correct.
A top view of the rst eight moves in a counterclockwise Towers of Hanoi solution
2
CS 473G
Final Exam Questions (December 13, 2005)
Fall 2005
You have 90 minutes to answer four of these questions.
Write your answers in the separate answer booklet.
You may take the question sheet with you when you leave.
Cherno Bounds: If X is the sum of independent indicator variables and = E[X ], then the
following inequalities hold for any > 0:
Pr[X < (1 )] <
e
(1 )1
Pr[X > (1 + )] <
e
(1 + )1+
1. Describe and analyze an algorithm that randomly shues an array X [1 .. n], so that each of
the n! possible permutations is equally likely, in O(n) time. (Assume that the subroutine
Random(m) returns an integer chosen uniformly at random from the set {1, 2, , . . . , m} in
O(1) time.)
2. Let G be an undirected graph with weighted edges. A heavy Hamiltonian cycle is a cycle C
that passes through each vertex of G exactly once, such that the total weight of the edges
in C is at least half of the total weight of all edges in G. Prove that deciding whether a graph
has a heavy Hamiltonian cycle is NP-complete.
5
2
8
1
6
4
5
9
12
7
8
3
5
A heavy Hamiltonian cycle. The cycle has total weight 34; the graph has total weight 67.
3. A sequence of numbers a1 , a2 , a3 , . . . an is oscillating if ai < ai+1 for every odd index i and
ai > ai+1 for every even index i. Describe and analyze an ecient algorithm to compute the
longest oscillating subsequence in a sequence of n integers.
4. This problem asks you to how to eciently modify a maximum ow if one of the edge
capacities changes. Specically, you are given a directed graph G = (V, E ) with capacities
c : E Z + , and a maximum ow F : E Z from some vertex s to some other vertex t
Z
Z
in G. Describe and analyze ecient algorithms for the following operations:
(a) Increment(e) Increase the capacity of edge e by 1 and update the maximum ow F .
(b) Decrement(e) Decrease the capacity of edge e by 1 and update the maximum ow F .
Both of your algorithms should be signicantly faster than recomputing the maximum ow
from scratch.
1
CS 473G
Final Exam Questions (December 13, 2005)
Fall 2005
5.
6. Let G = (V, E ) be an undirected graph, each of whose vertices is colored either red, green,
or blue. An edge in G is boring if its endpoints have the same color, and interesting if its
endpoints have dierent colors. The most interesting 3-coloring is the 3-coloring with the
maximum number of interesting edges, or equivalently, with the fewest boring edges.
(a) Prove that it is NP-hard to compute the most interesting 3-coloring of a graph. [Hint:
There is a one-line proof. Use one of the NP-hard problems described in class.]
(b) Let zzz (G) denote the number of boring edges in the most interesting 3-coloring of a
100
graph G. Prove that it is NP-hard to approximate zzz (G) within a factor of 1010 .
[Hint: There is a one-line proof.]
(c) Let wow(G) denote the number of interesting edges in the most interesting 3-coloring
of G. Suppose we assign each vertex in G a random color from the set {red, green, blue}.
2
Prove that the expected number of interesting edges is at least 3 wow(G).
7.
2
CS 473G
Final Exam Questions (December 13, 2005)
Fall 2005
You have 180 minutes to answer six of these questions.
Write your answers in the separate answer booklet.
You may take the question sheet with you when you leave.
1. Describe and analyze an algorithm that randomly shues an array X [1 .. n], so that each of
the n! possible permutations is equally likely, in O(n) time. (Assume that the subroutine
Random(m) returns an integer chosen uniformly at random from the set {1, 2, , . . . , m} in
O(1) time.)
2. Let G be an undirected graph with weighted edges. A heavy Hamiltonian cycle is a cycle C
that passes through each vertex of G exactly once, such that the total weight of the edges
in C is at least half of the total weight of all edges in G. Prove that deciding whether a graph
has a heavy Hamiltonian cycle is NP-complete.
5
2
8
1
6
4
5
9
12
7
8
3
5
A heavy Hamiltonian cycle. The cycle has total weight 34; the graph has total weight 67.
3. Suppose you are given a directed graph G = (V, E ) with capacities c : E Z + and a
Z
maximum ow F : E Z from some vertex s to some other vertex t in G. Describe and
Z
analyze ecient algorithms for the following operations:
(a) Increment(e) Increase the capacity of edge e by 1 and update the maximum ow F .
(b) Decrement(e) Decrease the capacity of edge e by 1 and update the maximum ow F .
Both of your algorithms should be signicantly faster than recomputing the maximum ow
from scratch.
4. Suppose you are given an undirected graph G and two vertices s and t in G. Two paths
from s to t are vertex-disjoint if the only vertices they have in common are s and t. Describe
and analyze an ecient algorithm to compute the maximum number of vertex-disjoint paths
between s and t in G. [Hint: Reduce this to a more familiar problem on a suitable directed
graph G .]
1
CS 473G
Final Exam Questions (December 13, 2005)
Fall 2005
5. A sequence of numbers a1 , a2 , a3 , . . . an is oscillating if ai < ai+1 for every odd index i and
ai > ai+1 for every even index i. For example, the sequence 2, 7, 1, 8, 2, 8, 1, 8, 3 is oscillating.
Describe and analyze an ecient algorithm to compute the longest oscillating subsequence in
a sequence of n integers.
6. Let G = (V, E ) be an undirected graph, each of whose vertices is colored either red, green,
or blue. An edge in G is boring if its endpoints have the same color, and interesting if its
endpoints have dierent colors. The most interesting 3-coloring is the 3-coloring with the
maximum number of interesting edges, or equivalently, with the fewest boring edges. Computing the most interesting 3-coloring is NP-hard, because the standard 3-coloring problem
we saw in class is a special case.
(a) Let zzz (G) denote the number of boring edges in the most interesting 3-coloring of a
100
graph G. Prove that it is NP-hard to approximate zzz (G) within a factor of 1010 .
(b) Let wow(G) denote the number of interesting edges in the most interesting 3-coloring
of G. Suppose we assign each vertex in G a random color from the set {red, green, blue}.
2
Prove that the expected number of interesting edges is at least 3 wow(G).
7. Its time for the 3rd Quasi-Annual Champaign-Urbana Ice Motorcycle Demolition Derby
Race-O-Rama and Spaghetti Bake-O! The main event is a competition between two teams
of n motorcycles in a huge square ice-covered arena. All of the motorcycles have spiked
tires so that they can ride on the ice. Each motorcycle drags a long metal chain behind it.
Whenever a motorcycle runs over a chain, the chain gets caught in the tire spikes, and the
motorcycle crashes. Two motorcycles can also crash by running directly into each other. All
the motorcycle start simultaneously. Each motorcycle travels in a straight line at a constant
speed until it either crashes or reaches the opposite wallno turning, no braking, no speeding
up, no slowing down. The Vicious Abscissas start at the south wall of the arena and ride
directly north (vertically). Hells Ordinates start at the west wall of the arena and ride directly
east (horizontally). If any motorcycle completely crosses the arena, that riders entire team
wins the competition.
Hell's Ordinates
Describe and analyze an ecient algorithm to decide which team will win, given the starting
position and speed of each motorcycle.
The Vicious Abscissas
2