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Unformatted text preview: c imal enc oding, unary enc oding will grow the input s iz e ex ponentially . Therefore, the
reduc tion to generate the c orres ponding input for UNARYSSUM will require a number of s teps
that is ex ponential in the s iz e of the input, tak ing more than poly nomial time. The proof fails .
2.)
SU BSET À SU M is in P bec aus e the following poly nomial time algorithm bas ed on dy namic
programming s olv e the problem.
Input: (S; t) , where S = f x1 ; x2 ; :::; xkg
Variables : Integers i; j; l[0 ::k][0 ::t] , all are initializ ed to 0
l[0 ][0 ]← 1 ;
f or i = 1 to k do
f or j = 0 to t À xi do
if l[i À 1 ][j] = 1 then l[i][j + xi] ← 1 ;
od
od
if l[k][t] = 1 then accep t else reject;
7.17
As s ume that P = N P : Let B 2 P but B = ¶ and B = ÆÃ s uc h that there ex is t a s tring
=
=
waccep t 2 B and a s tring wrej ect 2 B . To s how B in P = N P is NPc omplete, let A be an
=
arbitrary language in P = N P . Therefore, A c an be dec ided by a poly nomial time dec ider M A .
Nex t, we s how a poly nomial time reduc tion M from A to B as follows .
M = ”On input w :
1. Run M A on w .
2. If M A ac c epts then output waccep t . If M A rejec ted then output wrej ect .”
Obv ious ly , there ex is ts a poly nomial time reduc tion from A to B . Henc e B is NPc omplete. 7.20
a.)
SP AT H is in P bec aus e the following TM M c an dec ide it in poly nomial time.
M = “On input ⟨G; a; b; k⟩ where G is a graph with n nodes , two whic h are a and b:
1. Mark node a 0
2. For eac h i from 0 to n :
a. If an edge (s; t) is found c onnec ting s mark ed i to an unmark ed node i , mark
node t with i + 1 .
3. If b is mark ed with a v alue of at mos t b , ac c ept. Otherwis e, rejec t.”
b.)
LP AT H 2 N P bec aus e a path from a to b c an be gues s ed by an NTM nondeterminis tic ally
and c hec k ed if its length is at leas t k in poly nomial time.
Nex t, we s how U H AM P AT H Ôp LP AT H by c ons truc ting a TM F that c omputes the
reduc tion f .
F = “On input ⟨G; a; b⟩ of U H AM P AT H , where G is a graph with two nodes a and b:
1. Let k be the number of nodes of G.
2. Output ⟨G; a; b; k⟩ as an ins...
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 Spring '13
 zhang

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