The Entropy of Backwards Analysis

Backwards analysis, first popularized by Seidel, is often the simplest most elegant way of analyzing a randomized algorithm. It applies to incremental algorithms where elements are added incrementally, following some random permutation, e.g., incremental Delauney triangulation of a pointset, where points are added one by one, and where we always maintain the Delauney triangulation of the points added thus far. For backwards analysis, we think of the permutation as generated backwards, implying that the $i$th point in the permutation is picked uniformly at random from the $i$ points not picked yet in the backwards direction. Backwards analysis has also been applied elegantly by Chan to the randomized linear time minimum spanning tree algorithm of Karger, Klein, and Tarjan. The question considered in this paper is how much randomness we need in order to trust the expected bounds obtained using backwards analysis, exactly and approximately. For the exact case, it turns out that a random permutation works if and only if it is minwise, that is, for any given subset, each element has the same chance of being first. Minwise permutations are known to have $\Theta(n)$ entropy, and this is then also what we need for exact backwards analysis. However, when it comes to approximation, the two concepts diverge dramatically. To get backwards analysis to hold within a factor $\alpha$, the random permutation needs entropy $\Omega(n/\alpha)$. This contrasts with minwise permutations, where it is known that a $1+\varepsilon$ approximation only needs $\Theta(\log (n/\varepsilon))$ entropy. Our negative result for backwards analysis essentially shows that it is as abstract as any analysis based on full randomness

Constructing Light Spanners Deterministically in Near-Linear Time

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [Shiri Chechik and Christian Wulff-Nilsen, 2018] improved the state-of-the-art for light spanners by constructing a (2k-1)(1+epsilon)-spanner with O(n^(1+1/k)) edges and O_epsilon(n^(1/k)) lightness. Soon after, Filtser and Solomon [Arnold Filtser and Shay Solomon, 2016] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn^(1+1/k)) (which is faster than [Shiri Chechik and Christian Wulff-Nilsen, 2018]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Omega_epsilon(kn^(1/k)), even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an O_epsilon(n^(2+1/k+epsilon\u27)) time spanner construction which achieves the state-of-the-art bounds. Our second result is an O_epsilon(m + n log n) time construction of a spanner with (2k-1)(1+epsilon) stretch, O(log k * n^(1+1/k) edges and O_epsilon(log k * n^(1/k)) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=log n, for every constant epsilon>0, we provide an O(m+n^(1+epsilon)) time construction that produces an O(log n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k = omega(1). To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest