44 Randomization and Derandomization

FRAMEWORK AND ANALYSIS Most randomized incremental algorithms in the literature follow the framework sketched here for the computation of the trapezoidal map: the structure to be computed is maintained while the objects defining it are inserted in random order. To insert a new object, one first has to find a “conflict” of that object (the location step), then local updates in the structure are sufficient to bring it up to date (the update step). The cost of the update is usually linear in the size of the change in the combinatorial structure being maintained, and can often be bounded using backwards analysis. The location step can be implemented using either a conflict graph or a history graph. In both cases, the analysis is the same (since the actual computations performed are also often identical). To avoid having to prove the same bounds repeatedly for different problems, researchers have defined an axiomatic framework that captures the combinatorial essence of most randomized incremental algorithms. This framework, which uses configuration spaces, provides ready-to-use bounds for the expected running time of most randomized incremental algorithms. See Section 44.5. POINT LOCATION THROUGH HISTORY GRAPH In our trapezoidal map example, the history graph may be used as a point location structure for the trapezoidal map: given a query point q, find the trapezoid containing q by following a path from the root to a leaf node of the history graph. At each step, we continue to the child node corresponding to the trapezoid containing q. The search time is clearly proportional to the length of the path. Backwards analysis shows that the expected length of this path is O(log n) for any fixed query Preliminary version (December 16, 2016). 1166 O. Cheong, K. Mulmuley, and E. Ramos point. Even stronger, one can show that the maximum length of any search path in the history graph is O(log n) with high probability. If point location is the goal, the history graph can be simplified: instead of storing trapezoids, the inner nodes of the graph can denote two different kinds of elementary tests (“Does a point lie to the left or right of another point?” and “Does a point lie above or below a line?”). The final result is then an efficient and practical planar point location structure [Sei91]. This observation can also lead to a somewhat different location step inside the randomized incremental algorithm. Instead of performing a graph search with the whole segment s, point location can be used to find the trapezoid containing one endpoint of s. From there, a traversal of the trapezoidal map allows locating all trapezoids intersected by s.