By R Cooke, K L. Lockett, J A Bellman

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The layout and research of geometric algorithms has visible awesome development in recent times, as a result of their program in laptop imaginative and prescient, snap shots, scientific imaging, and CAD. Geometric algorithms are outfitted on 3 pillars: geometric facts constructions, algorithmic info structuring strategies and effects from combinatorial geometry.

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**Example text**

The digital trees are used very extensively in this book as illustrative examples. We analyze the height of tries, PATRICIA tries, and suffix trees in Chapter 4. Recurrences arising in the analysis of digital trees are discussed in Chapter 7, while the typical depth of digital search trees is studied in Chapters 8-10. 2 DATA COMPRESSION: LEMPEL-ZIV ALGORITHMS Source coding is an area of information theory (see Chapter 6) that deals with problems of optimal data compression. The most successful and best-known data compression schemes are due to Lempel and Ziv [460,461].

1 Some Useful Inequalities We review here some inequalities that play a considerable role in probability theory; they are often used in the probabilistic analysis of algorithms. 6) Review of Probability 31 for any t > 0. Indeed, we have the following chain of obvious inequalities E[g(X)} > E[g(X)I(g(X) > r)] > fE[/(g(X) > t)] = tPr{g(X) > t], where we recall that 1(A) is the indicator function of the event A. 7) which is known as Chebyshev's inequality. Schwarz's Inequality (also called Cauchy-Schwarz) Let X and Y be such that E[X2] < oo and E[Y2] < oo.

Choose X, Y e I,Z = X®Y; 4. 5. /<-(/\{X,K})U{Z}; Off ^Of + o(X,Y); 6. until |/| = 1 Different variants of the above generic algorithm can be envisioned by interpreting appropriately the "choose" statement in Step 3 above: RGREEDY: In Step 3, X is the string Z produced in the previous iteration, while Y is chosen in order to maximize o(X, Y) = o(Z, Y). Our initial choice for X is X1. Thus, in RGREEDY we have one "long" string Z which grows by addition 20 Data Structures and Algorithms on Words of strings at the right hand end.