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We have already seen that spanning trees and cuts are intimately related Here is another connection Let s remove the last edge that Kruskal s algorithm adds to the spanning tree; this breaks the tree into two components, thus de ning a cut (S, S) in the graph What can we say about this cut Suppose the graph we were working with was unweighted, and that its edges were ordered uniformly at random for Kruskal s algorithm to process them Here is a remarkable fact: with probability at least 1/n 2 , (S, S) is the minimum cut in the graph, where the size of a cut (S, S) is the number of edges crossing between S and S This means that repeating the process O(n 2 ) times and outputting the smallest cut found yields the minimum cut in G with high probability: an O(mn 2 log n) algorithm for unweighted minimum cuts Some further tuning gives the O(n 2 log n) minimum cut algorithm, invented by David Karger, which is the fastest known algorithm for this important problem So let us see why the cut found in each iteration is the minimum cut with probability at least 1/n2 At any stage of Kruskal s algorithm, the vertex set V is partitioned into connected components The only edges eligible to be added to the tree have their two endpoints in distinct components The number of edges incident to each component must be at least C, the size of the minimum cut in G (since we could consider a cut that separated this component from the rest of the graph) So if there are k components in the graph, the number of eligible edges is at least kC/2 (each of the k components has at least C edges leading out of it, and we need to compensate for the double-counting of each edge) Since the edges were randomly ordered, the chance that the next eligible edge in the list is from the minimum cut is at most C/(kC/2) = 2/k Thus, with probability at least 1 2/k = (k 2)/k, the choice leaves the minimum cut intact But now the chance that Kruskal s algorithm leaves the minimum cut intact all the way up to the choice of the last spanning tree edge is at least n 2 n 3 n 4 2 1 1 = n n 1 n 2 4 3 n(n 1).

We start with 0, which is equal to by definition The numbers are built upon each other as sets within sets, like this: 0= 1 = {0} = { }

.

2 = {0, 1} = { , { }} 3 = {0, 1, 2} = { , { }, { , { }}} 4 = {0, 1, 2, 3} = { , { }, { , { }}, { , { }, { , { }}}}

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Let s return to our discussion of minimum spanning tree algorithms What the cut property tells us in most general terms is that any algorithm conforming to the following greedy schema is guaranteed to work X = { } (edges picked so far) repeat until |X| = |V | 1: pick a set S V for which X has no edges between S and V S let e E be the minimum-weight edge between S and V S X = X {e} A popular alternative to Kruskal s algorithm is Prim s, in which the intermediate set of edges X always forms a subtree, and S is chosen to be the set of this tree s vertices On each iteration, the subtree de ned by X grows by one edge, namely, the lightest edge between a vertex in S and a vertex outside S (Figure 58) We can equivalently think of S as 139

Question 3-3

close Some individual stocks and exchange-traded funds (ETFs) experienced much wilder swings For example, a few stocks went from $40 to a penny within minutes Flash traders, the likely culprits of the mysterious Flash Crash, use high-speed electronic computers and questionable strategies to peek at other people s orders microseconds ahead of everyone else Even though profits are usually a few pennies per share, all of these shares add up to billions of dollars per year After the minicrash, the NYSE and Nasdaq canceled approximately 19,000 trades made between 2:40 pm and 3:00 pm on that day But many other trades were left as is, leaving thousands of unlucky investors holding dramatically lower-priced securities According to reports, approximately 68 percent of the canceled trades included ETFs Buyers were nowhere to be found as other electronic traders simultaneously rushed toward the exits Unlike market makers and specialists, electronic traders are not obligated to provide a fair and orderly market In fact, it was also reported that many highfrequency traders turned off their computers, letting prices crash Other professional liquidity providers also did not participate on the buy side, intensifying the price declines To critics, flash trading sounds suspiciously like front running Many say that flash trading gives the owners an unfair edge over other market participants It allows an elite group special access to quotes and the ability to determine supply and demand Proponents, on the other hand, claim that flash trading adds liquidity to the market There is little doubt that flash trading gives an informational edge, generating billions of dollars in profits each year, but is it fair Although high-frequency trading using high-speed computers is probably here to stay, the controversial strategy of using flash orders may be banned or regulated in the future Until then, individual traders and investors may want to think twice before using market stop orders If you had used a market order on the day of the Flash Crash, instead of getting stopped out at $50, for example, your stop order might have been executed at $20 or lower.

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Answer 3-3

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