Unformatted text preview: CS 2110 Fall 2009 Prelim 2 Tuesday, November 16th (Tomorrow!) Uris G01 (same as Prelim 1) 7:30
9:00 Topics: Threads and Concurrency (basic) Big O ADTs: Stacks, Queues, PriorityQueues, Maps, Sets Graph Algorithms: Prim’s, Kruskal’s, Dijkstra’s No Induction Don’t spend your time memorizing Java APIs! If you want to use an ADT, it’s acceptable to write code that looks reasonable, even if it’s not the exact Java API. For example, Queue<Integer> myQueue = new Queue<Integer>(); myQueue.enqueue(5); … int x = myQueue.dequeue(); This is not correct Java (Queue is an interface! And Java calls enqueue and dequeue “add” and “poll”) But it’s ﬁne for the exam. Big
O is an asymptotic upper bound on a function “f(x) is O(g(x))” Meaning: There exists some constant k such that f(x) ≤ k g(x) …as x goes to inﬁnity Often used to describe upper bounds for both worst
case and average
case algorithm runtimes Runtime is a function: The number of operations performed, usually as a function of input size For the prelim, you should know… Worst case Big
O complexity for the algorithms we’ve covered and for common implementations of ADT operations ▪ Examples ▪ Mergesort is worst
case O(n log n) ▪ PriorityQueue insert using a heap is O(log n) Average case time complexity for some algorithms and ADT operations, if it has been noted in class ▪ Examples ▪ Quicksort is average case O(n log n) ▪ HashMap insert is average case O(1) For the prelim, you should know… How to estimate the Big
O worst case runtimes of basic algorithms (written in Java or pseudocode) ▪ Count the operations ▪ Loops tend to multiply the loop body operations by the loop counter ▪ Trees and divide
and
conquer algorithms tend to introduce log(n) as a factor in the complexity ▪ Basic recursive algorithms, i.e., binary search or mergesort What do we mean by “abstract”? Deﬁned in terms of operations that can be performed, not as a concrete structure ▪ Example: Priority Queue is an ADT, Heap is a concrete data structure For ADTs, we should know: Operations oﬀered, and when to use them Big
O complexity of these operations for standard implementations interface Bag<E> {
void insert(E obj);
E extract(); //extract some element
boolean isEmpty();
E peek(); // optional: return next
element without removing
} Examples: Queue, Stack, PriorityQueue 8 First
In
First
Out (FIFO) Objects come out of a queue in the same order they were inserted Linked List implementation insert(obj): O(1) ▪ Add object to tail of list ▪ Also called enqueue, add (Java) extract(): O(1) ▪ Remove object from head of list ▪ Also called dequeue, poll (Java) 9 Last
In
First
Out (LIFO) Objects come out of a queue in the opposite order they were inserted Linked List implementation insert(obj): O(1) ▪ Add object to tail of list ▪ Also called push (Java) extract(): O(1) ▪ Remove object from head of list ▪ Also called pop (Java) 10 Objects come out of a Priority Queue according to their priority Generalized By using diﬀerent priorities, can implement Stacks or Queues Heap implementation (as seen in lecture) insert(obj, priority): O(log n) ▪ insert object into heap with given priority ▪ Also called add (Java) extract(): O(log n) ▪ Remove and return top of heap (minimum priority element) ▪ Also called poll (Java) 11 Concrete Data Structure Balanced binary tree Obeys heap order invariant: Priority(child) ≥ Priority(parent) Operations insert(value, priority) extract() • Put the new element at the end of the array • If this violates heap order because it is smaller than its parent, swap it with its parent • Continue swapping it up until it ﬁnds its rightful place • The heap invariant is maintained! 13 4 6 14 21 22 8 38 55 19 10 35 20 14 4 6 14 21 22 8 38 55 19 10 20 35 5 15 4 6 14 21 22 8 38 55 35 5 10 20 19 16 4 6 5 21 22 8 38 55 35 14 10 20 19 17 4 6 5 21 22 8 38 55 35 14 10 20 19 18 • Time is O(log n), since the tree is balanced – size of tree is exponential as a function of depth – depth of tree is logarithmic as a function of size 19 • Remove the least element – it is at the root • This leaves a hole at the root – ﬁll it in with the last element of the array • If this violates heap order because the root element is too big, swap it down with the smaller of its children • Continue swapping it down until it ﬁnds its rightful place • The heap invariant is maintained! 20 4 6 5 21 22 8 38 55 14 10 20 35 19 21 4 6 5 21 22 8 38 55 14 10 20 35 19 22 4 6 5 21 22 8 38 55 14 10 20 35 19 23 4 19 6 5 21 22 8 38 55 14 10 35 20 24 5 4 6 19 21 22 8 38 55 14 10 35 20 25 5 4 6 14 21 22 8 38 55 19 10 35 20 26 5 4 6 14 21 22 8 38 55 19 10 35 20 27 4 5 6 14 21 22 8 38 55 19 10 35 20 28 4 5 6 14 21 22 8 38 55 19 10 35 20 29 20 4 5 6 14 21 22 8 38 55 19 35 10 30 6 4 5 14 20 21 22 8 38 55 19 35 10 31 6 4 5 8 14 21 22 20 38 55 19 35 10 32 6 4 5 8 14 21 22 10 38 55 19 35 20 33 6 4 5 8 14 21 22 10 38 55 19 35 20 34 • Time is O(log n), since the tree is balanced 35 • Elements of the heap are stored in the array in order, going across each level from left to right, top to bottom • The children of the node at array index n are found at 2n + 1 and 2n + 2 • The parent of node n is found at (n – 1)/2 36 ADT Set Operations: ▪ void insert(Object element);
▪ boolean contains(Object element);
▪ void remove(Object element);
▪ int size();
▪ iteration No duplicates allowed Hash table implementation: O(1) insert and contains SortedSet tree implementation: O(log n) insert and contains A set makes no promises about ordering, but you can still iterate over it. 37 ADT Dictionary (aka Map) Operations: ▪ void insert(Object
▪ void update(Object
▪ Object find(Object
▪ void remove(Object
▪ boolean isEmpty();
▪ void clear(); key, Object value);
key, Object value);
key);
key); Think of: key = word; value = deﬁnition Where used: Symbol tables Wide use within other algorithms A HashMap is a particular implementation of the Map interface 38 Hash table implementation: Use a hash function to compute hashes of keys Store values in an array, indexed by key hash A collision occurs when two keys have the same hash How to handle collisions? ▪ Store another data structure, such as a linked list, in the array location for each key ▪ Put (key, value) pairs into that data structure insert and ﬁnd are O(1) when there are no collisions ▪ Expected complexity Worst case, every hash is a collision ▪ Complexity for insert and ﬁnd comes from the tertiary data structure’s complexity, e.g., O(n) for a linked list A HashMap is a particular implementation of the Map interface 39 Set of vertices (or nodes) V, set of edges E Number of vertices n = V Number of edges m = E Upper bound O(n2) on number of edges ▪ A complete graph has m = n(n
1)/2 Directed or undirected Directed edges have distinct head and tail Weighted edges Cycles and paths Connected components DAGs Degree of a node (in
and out
degree for directed graphs) You should be able to write a Vertex class in Java and implement standard graph algorithms using this class However, understanding the algorithms is much more important than memorizing their code A spanning tree is a subgraph of an undirected graph that: Is a tree Contains every vertex in the graph Number of edges in a tree m = n
1 Spanning tree with minimum sum edge weights Prim’s algorithm Kruskal’s algorithm Not necessarily unique Graph search algorithm, builds up a spanning tree from one root vertex Like BFS, but it uses a priority queue ▪ Priority is the weight of the edge to the vertex ▪ Also need to keep track of which edge we used Always picks smallest edge to an unvisited vertex Runtime is O(m log m) O(m) Priority Queue operations at log(m) each Prim’s Algorithm Example This is our original weighted graph. The numbers near the edges indicate their weight. Prim’s Algorithm Example Vertex D has been arbitrarily chosen as a starting point. Vertices A, B, E and F are connected to D through a single edge. A is the vertex nearest to D and will be chosen as the second vertex along with the edge AD. Prim’s Algorithm Example The next vertex chosen is the vertex nearest to either D or A. B is 9 away from D and 7 away from A, E is 15, and F is 6. F is the smallest distance away, so we highlight the vertex F and the arc DF. Prim’s Algorithm Example The algorithm carries on as above. Vertex B, which is 7 away from A, is highlighted. Prim’s Algorithm Example End Result Notice how each vertex has at least 1 edge connecting to it and that the edge is the least of the edges connected to the vertex. Idea: Find MST by connecting forest components using shortest edges Process edges from least to greatest Initially, every node is its own component Either an edge connects two diﬀerent components or it connects a component to itself ▪ Add an edge only in the former case Picks smallest edge between two components O(m log m) time to sort the edges ▪ Also need the union
ﬁnd structure to keep track of components, but it does not change the running time Kruskal’s Algorithm Example This is our original graph. The numbers near the arcs indicate their weight. None of the arcs are highlighted. ∞ Kruskal’s Algorithm Example AD and CE are the shortest arcs, with length 5, and AD has been arbitrarily chosen, so it is highlighted. Kruskal’s Algorithm Example CE is now the shortest arc that does not form a cycle, with length 5, so it is highlighted as the second arc. Kruskal’s Algorithm Example The next arc, DF with length 6, is highlighted using much the same method. Kruskal’s Algorithm Example The next
shortest arcs are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The arc BD has been highlighted in red, because there already exists a path (in green) between B and D, so it would form a cycle (ABD) if it were chosen. Kruskal’s Algorithm Example The process continues to highlight the next
smallest arc, BE with length 7. Many more arcs are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD. Kruskal’s Algorithm Example Finally, the process ﬁnishes with the arc EG of length 9, and the minimum spanning tree is found. Compute length of shortest path from source vertex to every other vertex Works on directed and undirected graphs Works only on graphs with non
negative edge weights O(m log m) runtime when implemented with Priority Queue, same as Prim’s Similar to Prim’s algorithm Diﬀerence lies in the priority Priority is the length of shortest path to a visited vertex + cost of edge to unvisited vertex We know the shortest path to every visited vertex On unweighted graphs, BFS gives us the same result as Dijkstra’s algorithm 1.
2.
3. 4. 5. Assign to every node a distance value. Set it to zero for our initial node and to inﬁnity for all other nodes. Mark all nodes as unvisited. Set initial node as current. For current node, consider all its unvisited neighbors and calculate their tentative distance (from the initial node) If this distance is less than the previously recorded distance, overwrite the distance. When we are done considering all neighbors of the current node, mark it as visited. A visited node will not be checked ever again; its distance recorded now is ﬁnal and minimal. If all nodes have been visited, ﬁnish. Otherwise, set the unvisited node with the smallest distance (from the initial node) as the next "current node" and continue from step 3. Dijkstra’s Algorithm Example 0 Initial distances set to 0 for initial node and ∞ for all other nodes. ∞ ∞ ∞ ∞ ∞ ∞ Dijkstra’s Algorithm Example 0 Set distances for all nodes connected to the initial node. Mark the initial node as done (red). ∞ 7 ∞ 5 ∞ ∞ Dijkstra’s Algorithm Example 0 Select the node is with the smallest distance that isn’t done, and update the distances to its neighbors. F = 11 : 5 + 6 = 11 B = 7: 5 + 9 = 14 > 7 E = 20: 5 + 15 = 20 Mark D as visited. ∞ 7 20 5 11 ∞ Dijkstra’s Algorithm Example 0 Set the current node to B. E = 14: 7 + 7 = 14 C = 15: 7 + 8 = 15 Mark B as visited. 15 7 14 5 11 ∞ Dijkstra’s Algorithm Example 0 Repeat the process: E = 14: 11 + 8 = 19 > 14 G = 22: 11 + 11 = 22 Mark F as visited 15 7 14 5 11 22 Dijkstra’s Algorithm Example 0 Repeat the process: C = 15: 14 + 5 = 19 > 15 G = 22: 14 + 9 = 23 > 22 Mark E as visited 15 7 14 5 11 22 Now we’ll take a 5
10 minute break We’ll begin Q&A session afterwards ...
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 Fall '07
 FRANCIS
 Algorithms, Dijkstra, the00

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