Brent Heeringa

Approximating Optimal Binary Decision Trees

We give a (ln n+1)-approximation for the decision tree (DT) problem. An instance of DT is a set of m binary tests T=(T1,…,Tm) and a set of n items X=(X1,…,Xn). The goal is to output a binary tree where each internal node is a test, each leaf is an item and the total external path length of the tree is minimized. Total external path length is the sum of the depths of all the leaves in the tree. DT has a long history in computer science with applications ranging from medical diagnosis to experiment design. It also generalizes the problem of finding optimal average-case search strategies in partially ordered sets which includes several alphabetic tree problems. Our work decreases the previous best upper bound on the approximation ratio by a constant factor. We provide a new analysis of the greedy algorithm that uses a simple accounting scheme to spread the cost of a tree among pairs of items split at a particular node. We conclude by showing that our upper bound also holds for the DT problem with weighted tests.

Hierarchical agent control: a framework for defining agent behavior

The Hierarchical Agent Control Architecture (HAC) is a general toolkit for specifying an agents behavior. HAC supports action abstraction, resource management, sensor integration, and is well suited to controlling large numbers of agents in dynamic environments. It relies on three hierarchies: action, sensor, and context. The action hierarchy controls the agents behavior. It is organized around tasks to be accomplished, not the agents themselves. This facilitates the integration of multi-agent actions and planning into the architecture. The sensor hierarchy provides a principled means for structuring the complexity of reading and transforming sensor information. Each level of the hierarchy integrates the data coming in from the environment into conceptual chunks appropriate for use by actions at this level. Actions and sensors are written using the same formalism. The context hierarchy is a hierarchy of goals. In addition to their primary goals, most actions are operating within a set of implicit assumptions. These assumptions are made explicit through the context hierarchy. We have developed a planner, GRASP, implemented within HAC, which is capable of resolving multiple goals in real time. HAC was intended to have wide applicability. It has been used to control agents in commercial computer games and physical robots. Our primary application domain is a simulator of land-based military engagements called “Capture the Flag.” HACs simulation substrate models physics at an abstract level. HAC supports any domain in which behaviors can be reduced to a small set of primitive effectors such as {\sc move} and {\sc apply-force}. At this time defining agent behavior requires Lisp programming skills; we are moving towards more graphical programming languages.

Approximating minimum reset sequences

We consider the problem of finding minimum reset sequences in synchronizing automata. The well-known Cerný conjecture states that every n-state synchronizing automaton has a reset sequence with length at most (n - 1)2. While this conjecture gives an upper bound on the length of every reset sequence, it does not directly address the problem of finding the shortest reset sequence. We call this the MINIMUM RESET SEQUENCE (MRS) problem. We give an O(kmnk + n4/k)-time ⌈n-1/k-1⌉-approximation for the MRS problem for any k ≥ 2. We also show that our analysis is tight. When k = 2 our algorithm reduces to Eppsteins algorithm and yields an (n-1)-approximation. When k = n our algorithm is the familiar exponential-time, exact algorithm. We define a nontrivial class of MRS which we call STACK COVER. We show that STACK COVER naturally generalizes two classic optimization problems: MIN SET COVER and SHORTEST COMMON SUPERSEQUENCE. Both these problems are known to be hard to approximate, although at present, SET COVER has a slightly stronger lower bound. In particular, it is NP-hard to approximate SET COVER to within a factor of c ċ log n for some c > 0. Thus, the MINIMUM RESET SEQUENCE problem is as least as hard to approximate as SET COVER. This improves the previous best lower bound which showed that it was NP-hard to approximate the MRS on binary alphabets to within any constant factor. Our result requires an alphabet of arbitrary size.

The knapsack problem with neighbour constraints

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the vertices have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.

Estimating Grammar Parameters Using Bounded Memory

Estimating the parameters of stochastic context-free grammars (SCFGs) from data is an important, well-studied problem. Almost without exception, existing approaches make repeated passes over the training data. The memory requirements of such algorithms are ill-suited for embedded agents exposed to large amounts of training data over long periods of time. We present a novel algorithm, called HOLA, for estimating the parameters of SCFGs that computes summary statistics for each string as it is observed and then discards the string. The memory used by HOLA is bounded by the size of the grammar, not by the amount of training data. Empirical results show that HOLA performs as well as the Inside-Outside algorithm on a variety of standard problems, despite the fact that it has access to much less information.

Search space reductions for nearest-neighbor queries

The vast number of applications featuring multimedia and geometric data has made the R-tree a ubiquitous data structure in databases. A popular and fundamental operation on R-trees is nearest neighbor search. While nearest neighbor on R-trees has received considerable experimental attention, it has received somewhat less theoretical consideration. We study pruning heuristics for nearest neighbor queries on R-trees. Our primary result is the construction of non-trivial families of R-trees where k-nearest neighbor queries based on pessimistic (i.e. min-max) distance estimates provide exponential speedup over queries based solely on optimistic (i.e. min) distance estimates. The exponential speedup holds even when k = 1. This result provides strong theoretical evidence that min-max distance heuristics are an essential component to depth-first nearest-neighbor queries. In light of this, we also consider the time-space tradeoffs of depth-first versus best-first nearest neighbor queries and construct a family of R-trees where best-first search performs exponentially better than depth-first search even when depth-first employs min-max distance heuristics.

Optimal Website Design with the Constrained Subtree Selection Problem

We introduce the Constrained Subtree Selection (CSS) problem as a model for the optimal design of websites. Given a hierarchy of topics represented as a DAG G and a probability distribution over the topics, we select a subtree of the transitive closure of G which minimizes the expected path cost. We define path cost as the sum of the page costs along a path from the root to a leaf. Page cost, γ, is a function of the number of links on a page. We give a sufficient condition for γ which makes CSS NP-Complete. This result holds even for the uniform probability distribution. We give a polynomial time algorithm for instances of CSS where G does not constrain the choice of subtrees and γ favors pages with at most k links. We show that CSS remains NP-Hard for constant degree DAGs, but also provide an O(log(k)γ(d+1)) approximation for any G with maximum degree d, provided that γ favors pages with at most k links. We also give a complete characterization of the optimal trees for two special cases: (1) linear degree cost in unconstrained graphs and uniform probability distributions, and (2) logarithmic degree cost in arbitrary DAGs and uniform probability distributions.

Searching in dynamic tree-like partial orders

We give the first data structure for the problem of maintaining a dynamic set of n elements drawn from a partially ordered universe described by a tree. We define the LINE-LEAF Tree, a linear-sized data structure that supports the operations: insert; delete; test membership; and predecessor. The performance of our data structure is within an O(log w)-factor of optimal. Here w ≤ n is the width of the partial-order--a natural obstacle in searching a partial order.

The 1-neighbour knapsack problem

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.