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**11 - 13**of**13**### Persistence, Offline Algorithms, and Space Compaction

, 1991

"... We consider dynamic data structures in which updates rebuild a static solution. Space bounds for persistent versions of these structures can often be reduced by using an offline persistent data structure in place of the static solution. We apply this technique to decomposable search problems, to dyn ..."

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We consider dynamic data structures in which updates rebuild a static solution. Space bounds for persistent versions of these structures can often be reduced by using an offline persistent data structure in place of the static solution. We apply this technique to decomposable search problems, to dynamic linear programming, and to maintaining the minimum spanning tree in a dynamic graph. Our algorithms admit trade-offs of update time vs. query time, and of time vs. space.

### Thesis Summary The Diameter of Permutation Groups Fully Persistent Search Trees

, 1986

"... This thesis comprise two disjoint topics: the diameter of permutation groups and fully persistent search trees. The diameter of a permutation group is the length of the longest product of generators required to reach a group element. For example, the diameter of a permutation group puzzle like Rubik ..."

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This thesis comprise two disjoint topics: the diameter of permutation groups and fully persistent search trees. The diameter of a permutation group is the length of the longest product of generators required to reach a group element. For example, the diameter of a permutation group puzzle like Rubik's Cube is the.,largest number of moves necessary to solve the puzzle. There are well known polynomial-time algorithims to determine if it is possible to reach a particular permutation with a given set of generators, but these algorithms can give a product exponentially longer than is required. We show that if the generators are constrained to be cycles with degree bounded by a constant then the diameter of the group is O(n2). Moreover, an O(n 2) length product expressing a given permutation can be found in polynomial time. A persistent search tree differs from an ordinary search tree in that after an insertion or deletion, the old version of the tree can still be searched. This thesis will describe lazy evaluation techniques for search trees that allow them to be made fully persistent. A fully persistent search tree supports insertions, deletions, and queries in any version, past or present. The time per query or update is O(log m) where m is the total number of updates, and the space needed is O(1) per update. These bounds are the best possible. Contents