Input s t where s f x1 x2 xkg variables integers

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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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