高级算法 (Fall 2024)/Problem Set 1

• 每道题目的解答都要有完整的解题过程，中英文不限。

• 我们推荐大家使用LaTeX, markdown等对作业进行排版。

Problem 1 (Min-cut/Max-cut)

For any [math]\displaystyle{ \alpha \ge 1 }[/math], a cut is called an [math]\displaystyle{ \alpha }[/math]-approximate min-cut in a multigraph [math]\displaystyle{ G }[/math] if the number of edges in it is at most [math]\displaystyle{ \alpha }[/math] times that of the min-cut. Prove that the number of [math]\displaystyle{ \alpha }[/math]-approximate min-cuts in a multigraph [math]\displaystyle{ G }[/math] is at most [math]\displaystyle{ n^{2\alpha} / 2 }[/math]. (Hint: Run Karger's algorithm until it has [math]\displaystyle{ \lceil 2\alpha \rceil }[/math] supernodes. What is the chance that a particular [math]\displaystyle{ \alpha }[/math]-approximate min-cut is still available? How many possible cuts does this collapsed graph have?)

Problem 2 (Fingerprinting)

Two rooted trees [math]\displaystyle{ T_1 }[/math] and [math]\displaystyle{ T_2 }[/math] are said to be isomorphic if there exists a one to one mapping [math]\displaystyle{ f }[/math] from the nodes of [math]\displaystyle{ T_1 }[/math] to those of [math]\displaystyle{ T_2 }[/math] satisfying the following condition: [math]\displaystyle{ v }[/math] is a child of [math]\displaystyle{ w }[/math] in [math]\displaystyle{ T_1 }[/math] if and only if [math]\displaystyle{ f(v) }[/math] is a child of [math]\displaystyle{ f(w) }[/math] in [math]\displaystyle{ T_2 }[/math]. Observe that no ordering is assumed on the children of any vertex. Devise an efficient randomized algorithm for testing the isomorphism of rooted trees and analyze its performance. Hint: Recursively associate a polynomial [math]\displaystyle{ P_v }[/math] with each vertex [math]\displaystyle{ v }[/math] in a tree [math]\displaystyle{ T }[/math].