Christian Konrad

Maximum Matching in Turnstile Streams

We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass 2-approximation streaming algorithm can be easily obtained with space O(n logn), where n denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space O(n3/2 − δ) is granted, for every δ > 0. Specifically, for every 0 ≤ e ≤ 1, we show that in the one-pass turnstile streaming model, in order to compute a O(n e )-approximation, space Ω(n3/2 − 4e) is required for constant error randomized algorithms, and, up to logarithmic factors, space \(\tilde{\mathrm{O}}( n^{2-2\epsilon} )\) is sufficient.

Validating XML documents in the streaming model with external memory

We study the problem of validating XML documents of size N against general DTDs in the context of streaming algorithms. The starting point of this work is a well-known space lower bound. There are XML documents and DTDs for which p-pass streaming algorithms require Ω(N/p) space. We show that when allowing access to external memory, there is a deterministic streaming algorithm that solves this problem with memory space O(log2 N), a constant number of auxiliary read/write streams, and O(log N) total number of passes on the XML document and auxiliary streams. An important intermediate step of this algorithm is the computation of the First-Child-Next-Sibling (FCNS) encoding of the initial XML document in a streaming fashion. We study this problem independently, and we also provide memory efficient streaming algorithms for decoding an XML document given in its FCNS encoding. Furthermore, validating XML documents encoding binary trees in the usual streaming model without external memory can be done with sublinear memory. There is a one-pass algorithm using O(√N log N) space, and a bidirectional two-pass algorithm using O(log2 N) space performing this task.

Approximating Semi-matchings in Streaming and in Two-Party Communication

We study the streaming complexity and communication complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph <i>G</i> = (<i>A</i>, <i>B</i>, <i>E</i>) with <i>n</i> = |<i>A</i>| is a subset of edges <i>S</i>⊆<i>E</i> that matches all <i>A</i> vertices to <i>B</i> vertices with the goal usually being to do this as fairly as possible. While the term <i>semi-matching</i> was coined in 2003 by Harvey et al. [2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 ⩽ ε ⩽ 1 uses space Õ(<i>n</i>^1+ε and computes an O(<i>n</i>^(1−ε)/2)-approximation to the semi-matching problem. Furthermore, with O(log <i>n</i>) passes it is possible to compute an O(log <i>n</i>)-approximation with space Õ(<i>n</i>). In the one-way two-party communication setting, we show that for every ε > 0, deterministic communication protocols for computing an O(<i>n</i>&frac;^{1(1+ε)<i>c</i>+1)}-approximation require a message of size more than <i>cn</i> bits. We present two deterministic protocols communicating <i>n</i> and 2<i>n</i> edges that compute an O&sqrt;<i>n</i> and an O(n^1/3)-approximation, respectively. Finally, we improve on the results of Harvey et al. [2003] and prove new links between semi-matchings and matchings. While it was known that an optimal semi-matching contains a maximum matching, we show that there is a hierarchical decomposition of an optimal semi-matching into maximum matchings. A similar result holds for semi-matchings that do not admit length-two degree-minimizing paths.

On the Power of Advice and Randomization for Online Bipartite Matching.

While randomized online algorithms have access to a sequence of uniform random bits, deterministic online algorithms with advice have access to a sequence of advice bits, i.e., bits that are set by an all powerful oracle prior to the processing of the request sequence. Advice bits are at least as helpful as random bits, but how helpful are they? In this work, we investigate the power of advice bits and random bits for online maximum bipartite matching (MBM). The well-known Karp-Vazirani-Vazirani algorithm is an optimal randomized

Distributed Algorithms for Coloring Interval Graphs

(1-\frac{1}{e})

Brief Announcement: Local Independent Set Approximation

-competitive algorithm for \textsc{MBM} that requires access to

Distributed Large Independent Sets in One Round on Bounded-Independence Graphs

\Theta(n \log n)

Robust set reconciliation

uniform random bits. We show that

The Minimum Vulnerability Problem on Graphs

\Omega(\log(\frac{1}{\epsilon}) n)

Streaming Partitioning of Sequences and Trees

advice bits are necessary and