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Reviewer: Joseph J. O'Rourke

Davenport-Schinzel sequences are deep and beautiful, simple but subtle combinatorial structures arising in a wide variety of geometric constructions. The remarkable story of their analysis is told in this book by two of the primary developers of the theory. The material is technically demanding and the style correspondingly uncompromising. The proofs are clearly presented, although often dauntingly intricate. A Davenport-Schinzel sequence (hereafter, a DS n,s sequence) is a string of symbols drawn from a set of n symbols. There are two restrictions on its structure. First, no two adjacent symbols are equal, which forces the symbols to alternate. The second restriction limits the alternation according to a parameter s : no alternating subsequence of length s+2 is permitted. Thus, for s=2 , any subsequence a..b..a..b is forbidden; for s=3,a..b..a..b..a is forbidden. These forbidden subsequences need not be contiguous. A DS n,s sequence cannot be too long for a fixed value of s ; let l s n be its maximum length. The first third of the book is devoted to careful and detailed proofs of bounds on this function. For example, l 2 3 =5 ; for instance, 12321 . It is not difficult to prove that l 2 n =2n-1 . The book illustrates how this bound determines the complexity of several geometric algorithms. Here is one example. The view of a collection of n labeled, opaque, vertical-axis parabolas from y=- (the “lower envelope” of the parabolas) is a DS n,2 sequence: the left-to-right list of arc labels could only contain the subsequence a..b..a..b if the a and b parabolas intersected at least three times. Fortunes sweepline algorithm for constructing Voronoi diagrams maintains a parabolic wavefront whose combinatorial complexity is l 2 n , and the linearity of this complexity is crucial in establishing the O n log n bound on the algorithm. Where the theory gets deep is for parameter values s>2 . Already l 3 n is exceedingly difficult to accurately bound. For n=2,3,4, l 3 n =4,8,12 ; for example, 12131323 realizes l 3 3 . One might expect linear growth, but l 3 5 =17 . In fact, l 3 n grows superlinearly, with the factor beyond linearity the inverse Ackermann function a n , which grows without bound but remains no more than 4 for n an exponential tower of 65,536 2s. This bound is asymptotically tight: the authors provide an explicit construction of a DS n,3 sequence of length W n a n . Moreover, and perhaps more surprisingly, this length can be achieved by the lower envelope of a collection of straight segments in the plane. “The construction is fairly involved,” to quote the authors understatement. One implication is that, under reasonable assumptions, no (one-dimensional) graphics image of interpenetrating segments can be computed with an algorithm of linear asymptotic complexity. Tight bounds have also been obtained for s=4: l 4 n = Q n2 a n , only slightly less gradual in its supralinearity. After carefully proving this and all known DS n,s bounds, the authors turn to the role they play in arrangements, partitions of space by collections of geometric objects. For example, they prove that the complexity of a single cell in an arrangement of segments is l 3 n . They detail a delicate divide-and-conquer algorithm to construct the single cell containing any specified point in near-linear time: O l 3 n log 2 n . In three dimensions, they establish the long-sought result that the complexity of the lower envelope of a collection of n surfaces is nearly quadratic: O n 2+ e for any e >0 ; the result generalizes to higher dimensions. The final chapter of the book details a wealth of applications, which include the following two examples. First, imagine you are riding a rocket shooting vertically above a polyhedral terrain of n edges. The number of combinatorial changes in the view you will see outside the window on your way to infinity is O n 22 a n ; examples are known that realize W n 2 a n . Second, the Voronoi diagram encodes proximity information about its point sites. If the points are moving, their proximity relations might change over time, reflected in structural changes in the Voronoi diagram. The number of these changes can be bounded by Davenport-Schinzel sequences. For n points moving in the plane along lines at the same speed, the total number of combinatorial changes in the diagram can be connected to the complexity of the lower envelope of trivariate functions, yielding an O n 3 bound. But the best lower bound known on Voronoi changes is W n 2 . Closing this gap is one of the many open problems that I hope will attract researchers, now that this impressive book has made the material accessible.