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1. The decision problem PARTITION is defined on page 13 of the Notes NP and NPCompleteness". (You may assume that a1; : : : am are positive integers.)

Define the associated search problem PARTITION-SEARCH and give an algorithm

showing that

PARTITION-SEARCH ! p PARTITION.

Give a loop invariant for your algorithm.

(See Definition 6 in the Notes Search and Optimization Problems" for the definition

of ! p .)

2. Consider the problem DISTANCE-PATH.

DISTANCE-PATH

Instance

hG; s; t; di, where G is an undirected graph, s and t are nodes in G, and d is a positive

integer.

Question Is the distance from s to t exactly d? In other words, is it the case that there

is a path of length d from s to t, and no shorter path from s to t?

(a) Show that DISTANCE-PATH 2 NL.

(b) Show that DISTANCE-PATH is NL-complete.

Hint: Show that P AT H ?L DISTANCE-PATH. Given a directed graph G construct

an undirected graph G0 by making n copies of G. Each edge in G0 goes from copy i to

copy i + 1.

3. Use a padding argument to show that NL = coNL implies NSPACE(n3) = coNSPACE(n3).

See Problem 9.13, in the textbook for a description of padding.

4. Show that T QBF = 2 DSPACE(n1=5). You may refer to the proof of Theorem 8.9 in

the text, and assume the fact that the reduction presented there can be carried out in

log space.

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