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Don’t Rush into a Union: Take Time to Find Your Roots
, 2011
"... We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times. Consider incremental connectivity, letting tU be the time to insert an edge and tq be the query time. For tU = Ω(tq), the problem is equivalent to the wellundersto ..."
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We present a new threshold phenomenon in data structure lower bounds where slightly reduced update times lead to exploding query times. Consider incremental connectivity, letting tU be the time to insert an edge and tq be the query time. For tU = Ω(tq), the problem is equivalent to the wellunderstood union–find problem: INSERTEDGE(s, t) can be implemented by UNION(FIND(s), FIND(t)). This gives worstcase time tU = tq = O(lg n / lg lg n) and amortized tU = tq = O(α(n)). By contrast, we show that if tU = o(lg n / lg lg n), the query time explodes to tq ≥ n 1−o(1). In other words, if the data structure doesn’t have time to find the roots of each disjoint set (tree) during edge insertion, there is no effective way to organize the information! For amortized complexity, we demonstrate a new inverseAckermann type tradeoff in the regime tU = o(tq). A similar lower bound is given for fully dynamic connectivity, where an update time of o(lg n) forces the query time to be n 1−o(1). This lower bound allows for amortization and Las Vegas randomization, and comes close to the known O(lg n · (lg lg n) O(1) ) upper bound. 1
Finding Path Minima in Incremental Unrooted Trees ∗
, 2008
"... Consider a dynamic forest of unrooted trees over a set of n vertices which we update by link operations: Each link operation adds a new edge adjacent to vertices in two different trees. Every edge in the forest has a weight associated with it, and at any time we want to be able to answer a pathmin ..."
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Consider a dynamic forest of unrooted trees over a set of n vertices which we update by link operations: Each link operation adds a new edge adjacent to vertices in two different trees. Every edge in the forest has a weight associated with it, and at any time we want to be able to answer a pathmin query which returns that edge of minimum weight along the path between two given vertices. For the case where the weights are integers we give an algorithm that performs n − 1 link operations and m pathmin queries in O(n + mα(m,n)) time. This extends well known results of Tarjan [11] and Yao [12] to a more general dynamic setting at the cost of restricting the weights to be integers. Using our data structure we get an optimal data structure for a restricted version of the mergeable trees problem [9]. We also suggest a simpler data structures for the case where trees are rooted and the link operation always adds an edge between the root of one tree and an arbitrary vertex of another tree.
Seminar on Advanced topics in data structures Fall 2000/2001
, 2000
"... We shall focus on the following three problems in data structures 1. The Unionfind problem. 2. Constructing optimal alphabetic binary trees. 3. The suffix tree. Depending upon the number of students participating we may touch other topics as well. ..."
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We shall focus on the following three problems in data structures 1. The Unionfind problem. 2. Constructing optimal alphabetic binary trees. 3. The suffix tree. Depending upon the number of students participating we may touch other topics as well.