algorithms

Recommendations

Info

This graph has multiple edges between two vertices—a feature we have not been allowing so far in this book, but one that is meaningful for this particular problem, since each bridge must be accounted for separately. We are looking for a path that goes through each edge exactly once (the path is allowed to repeat vertices). In other words, we are asking this question: When can a graph be drawn without lifting the pencil from the paper? The answer discovered by Euler is simple, elegant, and intuitive: If and only if (a) the graph is connected and (b) every vertex, with the possible exception of two vertices (the start and final vertices of the walk), has even degree (Exercise 3.26). This is why Königsberg’s park was impossible to traverse: all four vertices have odd degree. To put it in terms of our present concerns, let us define a search problem called EULER PATH: Given a graph, find a path that contains each edge exactly once. It follows from Euler’s observation, and a little more thinking, that this search problem can be solved in polynomial time. Almost a millennium before Euler’s fateful summer in East Prussia, a Kashmiri poet named Rudrata had asked this question: Can one visit all the squares of the chessboard, without repeating any square, in one long walk that ends at the starting square and at each step makes a legal knight move? This is again a graph problem: the graph now has 64 vertices, and two squares are joined by an edge if a knight can go from one to the other in a single move (that is, if their coordinates differ by 2 in one dimension and by 1 in the other). See Figure 8.2 for the portion of the graph corresponding to the upper left corner of the board. Can you find a knight’s tour on your chessboard? Figure 8.2 Knight’s moves on a corner of a chessboard. This is a different kind of search problem in graphs: we want a cycle that goes through all vertices (as opposed to all edges in Euler’s problem), without repeating any vertex. And there is no reason to stick to chessboards; this question can be asked of any graph. Let us define the RUDRATA CYCLE search problem to be the following: given a graph, find a cycle that visits each vertex exactly once—or report that no such cycle exists. 2 This problem is ominously 2 In the literature this problem is known as the Hamilton cycle problem, after the great Irish mathematician who rediscovered it in the 19th century. 238
Figure 8.3 What is the smallest cut in this graph? reminiscent of the TSP, and indeed no polynomial algorithm is known for it. There are two differences between the definitions of the Euler and Rudrata problems. The first is that Euler’s problem visits all edges while Rudrata’s visits all vertices. But there is also the issue that one of them demands a path while the other requires a cycle. Which of these differences accounts for the huge disparity in computational complexity between the two problems? It must be the first, because the second difference can be shown to be purely cosmetic. Indeed, define the RUDRATA PATH problem to be just like RUDRATA CYCLE, except that the goal is now to find a path that goes through each vertex exactly once. As we will soon see, there is a precise equivalence between the two versions of the Rudrata problem. Cuts and bisections A cut is a set of edges whose removal leaves a graph disconnected. It is often of interest to find small cuts, and the MINIMUM CUT problem is, given a graph and a budget b, to find a cut with at most b edges. For example, the smallest cut in Figure 8.3 is of size 3. This problem can be solved in polynomial time by n − 1 max-flow computations: give each edge a capacity of 1, and find the maximum flow between some fixed node and every single other node. The smallest such flow will correspond (via the max-flow min-cut theorem) to the smallest cut. Can you see why? We’ve also seen a very different, randomized algorithm for this problem (page 143). In many graphs, such as the one in Figure 8.3, the smallest cut leaves just a singleton vertex on one side—it consists of all edges adjacent to this vertex. Far more interesting are small cuts that partition the vertices of the graph into nearly equal-sized sets. More precisely, the BALANCED CUT problem is this: given a graph with n vertices and a budget b, partition the vertices into two sets S and T such that |S|, |T | ≥ n/3 and such that there are at most b edges between S and T . Another hard problem. Balanced cuts arise in a variety of important applications, such as clustering. Consider for example the problem of segmenting an image into its constituent components (say, an elephant standing in a grassy plain with a clear blue sky above). A good way of doing this is to create a graph with a node for each pixel of the image and to put an edge between nodes whose corresponding pixels are spatially close together and are also similar in color. A single object in the image (like the elephant, say) then corresponds to a set of highly connected vertices in the graph. A balanced cut is therefore likely to divide the pixels into two clusters without breaking apart any of the primary constituents of the image. The first cut might, for instance, separate the elephant on the one hand from the sky and from grass on the other. A 239
Page 1:
Algorithms Copyright c○2006 S. Da
Page 4 and 5:
4 Paths in graphs 109 4.1 Distances
Page 7 and 8:
List of boxes Bases and logs . . .
Page 9 and 10:
Preface This book evolved over the
Page 11 and 12:
Chapter 0 Prologue Look around you.
Page 13 and 14:
The first question is moot here, as
Page 15 and 16:
for addition, which works on one di
Page 17 and 18:
4. Likewise, any polynomial dominat
Page 19:
and in general ( Fn ) = F n+1 ( ) n
Page 22 and 23:
Bases and logs Naturally, there is
Page 24 and 25:
Figure 1.1 Multiplication à la Fra
Page 26 and 27:
Figure 1.3 Addition modulo 8. 6 0 0
Page 28 and 29:
Figure 1.4 Modular exponentiation.
Page 30 and 31:
Figure 1.6 A simple extension of Eu
Page 32 and 33:
We say x is the multiplicative inve
Page 34 and 35:
Figure 1.7 An algorithm for testing
Page 36 and 37:
Figure 1.8 An algorithm for testing
Page 38 and 39:
Randomized algorithms: a virtual ch
Page 40 and 41:
Alice’s encryption function is th
Page 42 and 43:
Figure 1.9 RSA. Bob chooses his pub
Page 44 and 45:
There is nothing inherently wrong w
Page 46 and 47:
Exercises 1.1. Show that in any bas
Page 48 and 49:
the depth of the circuit or the lon
Page 50 and 51:
(c) Give an example of positive int
Page 52 and 53:
x = x L x R = 2 n/2 x L + x R y = y
Page 54 and 55:
Figure 2.2 Divide-and-conquer integ
Page 56 and 57:
Binary search The ultimate divide-a
Page 59 and 60:
An n log n lower bound for sorting
Page 61 and 62:
The three sublists S L , S v , and
Page 63 and 64:
to be the best running time possibl
Page 66 and 67:
qp t AB CD EF GH IJ KL MN OP QR Why
Page 68 and 69:
The equivalence of the two polynomi
Page 70 and 71:
Figure 2.6 The complex roots of uni
Page 72 and 73:
A matrix reformulation To get a cle
Page 74 and 75:
The orthogonality property can be s
Page 76 and 77:
FFT n (input: a 0 , . . . , a n−1
Page 78 and 79:
The slow spread of a fast algorithm
Page 80 and 81:
(e) T (n) = 8T (n/2) + n 3 (f) T (n
Page 82 and 83:
2.21. Mean and median. One of the m
Page 84 and 85:
(c) In fact, squaring matrices is n
Page 87 and 88:
Chapter 3 Decompositions of graphs
Page 89 and 90:
Therefore, each edge appears in exa
Page 91 and 92:
Figure 3.3. 1 The previsit and post
Page 93 and 94:
Figure 3.6 (a) A 12-node graph. (b)
Page 95 and 96:
Tree edges are actually part of the
Page 97 and 98:
the same thing. Since a dag is line
Page 99 and 100:
Figure 3.10 The reverse of the grap
Page 101 and 102:
Exercises 3.1. Perform a depth-firs
Page 103 and 104:
(a) Model this as a graph problem:
Page 105 and 106:
For instance, in the graph below (w
Page 107 and 108:
3.30. On page 97, we defined the bi
Page 109 and 110:
Chapter 4 Paths in graphs 4.1 Dista
Page 111 and 112:
Figure 4.4 The result of breadth-fi
Page 113 and 114:
Figure 4.6 Breaking edges into unit
Page 115 and 116:
into account. This viewpoint gives
Page 117 and 118:
§¦ ¥¤ £¢ ©¨ #"
Page 119 and 120:
Which heap is best? The running tim
Page 121 and 122:
Figure 4.11 (a) A binary heap with
Page 123 and 124:
Figure 4.13 The Bellman-Ford algori
Page 125 and 126:
Figure 4.15 A single-source shortes
Page 127 and 128:
Input: Undirected graph G = (V, E)
Page 129 and 130:
4.16. Section 4.5.2 describes a way
Page 131 and 132:
Let r ∗ be the maximum ratio achi
Page 133 and 134:
Chapter 5 Greedy algorithms A game
Page 135 and 136:
Trees A tree is an undirected graph
Page 137 and 138:
Figure 5.3 The cut property at work
Page 139 and 140:
eturn x As can be expected, makeset
Page 141 and 142:
Figure 5.7 The effect of path compr
Page 143 and 144:
A randomized algorithm for minimum
Page 145 and 146:
Figure 5.9 Top: Prim’s minimum sp
Page 147 and 148:
Figure 5.10 A prefix-free encoding.
Page 149 and 150:
149
Page 151 and 152:
5.3 Horn formulas In order to displ
Page 153 and 154:
Figure 5.11 (a) Eleven towns. (b) T
Page 155 and 156:
Exercises 5.1. Consider the followi
Page 157 and 158:
(a) Code: {0, 10, 11} (b) Code: {0,
Page 159 and 160:
Input: Undirected graph G = (V, E);
Page 161 and 162:
Chapter 6 Dynamic programming In th
Page 163 and 164:
Figure 6.2 The dag of increasing su
Page 165 and 166:
Programming? The origin of the term
Page 167 and 168:
Figure 6.4 (a) The table of subprob
Page 169 and 170:
Common subproblems Finding the righ
Page 171 and 172:
6.4 Knapsack During a robbery, a bu
Page 173 and 174:
Memoization In dynamic programming,
Page 175 and 176:
Figure 6.7 (a) ((A × B) × C) × D
Page 177 and 178:
ecause it leads right back to the O
Page 179 and 180:
d ij . We must pick the best such i
Page 181 and 182:
Figure 6.11 I(u) is the size of the
Page 183 and 184:
6.8. Given two strings x = x 1 x 2
Page 185 and 186:
Figure 6.12 Two binary search trees
Page 187 and 188: 6.26. Sequence alignment. When a ne
Page 189 and 190: Chapter 7 Linear programming and re
Page 191 and 192: It is a general rule of linear prog
Page 193 and 194: the feasible region. The point of f
Page 195 and 196: Figure 7.3 A communications network
Page 197 and 198: Reductions Sometimes a computationa
Page 199 and 200: Matrix-vector notation A linear fun
Page 201 and 202: Figure 7.5 An illustration of the m
Page 203 and 204: L e R s t e ′ We claim that size(
Page 205 and 206: Figure 7.6 Continued Current Flow R
Page 207 and 208: from the algorithm, which in such c
Page 209 and 210: Figure 7.10 A generic primal LP in
Page 211 and 212: Now suppose the two of them play th
Page 213 and 214: In LP form, this is min w −3y 1 +
Page 215 and 216: 7.6.2 The algorithm On each iterati
Page 217 and 218: Figure 7.13 Simplex in action. Init
Page 219 and 220: close to one another? Unboundedness
Page 221 and 222: Gaussian elimination Under our alge
Page 223 and 224: output AND NOT OR AND OR NOT true f
Page 225 and 226: Exercises 7.1. Consider the followi
Page 227 and 228: 7.12. For the linear program max x
Page 229 and 230: • f (i) is a valid flow from s i
Page 231 and 232: 7.28. A linear program for shortest
Page 233 and 234: Chapter 8 NP-complete problems 8.1
Page 235 and 236: Satisfiability SATISFIABILITY, or S
Page 237: cost as the budget. Fine, but how d
Page 241 and 242: Figure 8.4 A more elaborate matchma
Page 243 and 244: 8.2 NP-complete problems Hard probl
Page 245 and 246: I. These two translation procedures
Page 247 and 248: Figure 8.7 Reductions between searc
Page 249 and 250: Figure 8.8 The graph corresponding
Page 251 and 252: Figure 8.9 S is a vertex cover if a
Page 253 and 254: 2n − m pets will be left unmatche
Page 255 and 256: Figure 8.10 Rudrata cycle with pair
Page 257 and 258: est of the graph as shown in Figure
Page 259 and 260: RUDRATA CYCLE−→TSP Given a grap
Page 261 and 262: Now that we know CIRCUIT SAT reduce
Page 263 and 264: Exercises 8.1. Optimization versus
Page 265 and 266: Input: An undirected graph G = (V,
Page 267 and 268: • (w i , w i ′ ) for all i = 1,
Page 269 and 270: Chapter 9 Coping with NP-completene
Page 271 and 272: anching. We can expand either of th
Page 273 and 274: follow the same pattern. As before,
Page 275 and 276: 9.2 Approximation algorithms In an
Page 277 and 278: 2. It can be used to build a vertex
Page 279 and 280: 9.2.3 TSP The triangle inequality p
Page 281 and 282: Let’s consider the O(nV ) algorit
Page 283 and 284: iterations will be needed: whether
Page 285 and 286: Figure 9.8 Local search. Figure 9.7
Page 287 and 288: What can be done about such subopti
Page 289 and 290:
Figure 9.10 The search space for a
Page 291 and 292:
Exercises 9.1. In the backtracking
Page 293 and 294:
start with any S ⊆ V while there
Page 295 and 296:
Chapter 10 Quantum algorithms This
Page 297 and 298:
Figure 10.2 Measurement of a superp
Page 299 and 300:
Figure 10.3 A quantum algorithm tak
Page 301 and 302:
¡ £¢ ¥¤ §¦ ©¨ #" %
Page 303 and 304:
The Fourier transform of a periodic
Page 305 and 306:
Yet another basic gate, the control
Page 307 and 308:
10.6 Factoring as periodicity We ha
Page 309 and 310:
Figure 10.6 Quantum factoring. 0 QF
Page 311 and 312:
Exercises 10.1. ∣ ψ 〉 = 1 √2
Page 313 and 314:
Historical notes and further readin
Page 315 and 316:
Index O(·), 13 Ω(·), 14 Θ(·),
Page 317 and 318:
interior-point method, 217 interpol
show all

algorithms

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?