## Description

1. Consider an undirected graph that weights of edges are distinct. We can define an order on the set of

spanning trees of the graph as follows: each spanning tree has n 1 edges, by sorting the weights of all

n 1 tree edges in the increasing order, we obtain a tuple (w1,…, wn1) where w1 < w2 < … < wn1. We can easily define a lexicographic order on tuples by imagining that each tuple is a word with n 1 letter. More precisely, we say tuple (w0 1,…, w0 n1) is bigger than tuple (w1,…, wn1) if for the smallest index i at which w0 i 6= wi, we have w0 i > wi. We say tree T1 > T2 if T1’s tuple is lexicographically bigger than T2’s

tuple. We define the lexicographically minimum tree to be the smallest tree in this ordering of trees. Prove

that the lexicographically minimum tree is a minimum spanning tree.

2. Given an undirected graph with distinct weights. Consider a spanning tree for the graph: given any pair of

vertices u and v, the tree can be used to return a path between u and v such that maximum-weight edge

on the path has a smaller weight than the maximum-weight edge of any other path between u and v.

(a) Prove that the spanning tree with the above property is the minimum spanning tree (mst).

(b) We define a order between two paths A and B. We sort the weights of edges in the decreasing order and

A = {w1, w2,…, wi}, B = {w0

1, w0

2,…, w0

j

}. We say A is bottleneck-smaller than B, if the first different

weight edge in A has smaller weight than the edge in B, more precisely if for the smallest index k

in A at which w0

k 6= wk, we have wk<w0

k. If there is not such k that w0

k 6= wk, the path with smaller

number of edges is bottleneck-smaller. We define the minimum bottleneck path as the path that is

bottleneck-smaller than any other path.

Consider the following variation of Dijkstra algorithm. Instead of computing the distance, we calculate

the minimum bottleneck path from the source s to any other node. Each node v maintain a variable

P(v) that stores the minimum bottleneck path from the s and ends at v. Moreover, the operation on

the priority queue is base on the order of P(v).

DIJKSTRAMST(s)

put s in the priority queue

T {}

while the priority queue is not empty

extract node u from the priority queue with the minimum bottleneck path

remove u from priority queue

T T [ {u}

for all edge u v that v is not in T

if v is not in priority queue

P(v) {P(v), u v}

put v in the priority queue

else if P0 = {P(u), u v} is bottleneck-smaller than P(v)

P(v) {P(v), u v}

i. Assume that in the execution of the DIJKSTRAMST you maintain a set contains all edges in all P(v).

Prove that at the end of the execution, such set defines a spanning tree of the graph.

ii. Prove that the spanning tree defined by DIJKSTRAMST is the mst.

iii. Assume that you execute DIJKSTRAMST and the Prim’s algorithm separately on node s. Prove or

disprove that the two algorithms construct the mst in the same order.

3. There is a multi-day tennis tournament with n = 2k players. On a given day, each player plays one match,

so n/2 matches are played per day. Over the course of the tournament, each player plays against every

other player exactly once, so a total of n(n 1)/2 matches will be played.

1

Write a recursive algorithm to schedule these matches in the minimum number of days. It should take as

input a number n = 2k and return as output a table with n 1 rows, in which row i is a list of the n/2

matches to be played on day i.

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