Philip N. Klein

Recognition of shapes by editing their shock graphs

This paper presents a novel framework for the recognition of objects based on their silhouettes. The main idea is to measure the distance between two shapes as the minimum extent of deformation necessary for one shape to match the other. Since the space of deformations is very high-dimensional, three steps are taken to make the search practical: 1) define an equivalence class for shapes based on shock-graph topology, 2) define an equivalence class for deformation paths based on shock-graph transitions, and 3) avoid complexity-increasing deformation paths by moving toward shock-graph degeneracy. Despite these steps, which tremendously reduce the search requirement, there still remain numerous deformation paths to consider. To that end, we employ an edit-distance algorithm for shock graphs that finds the optimal deformation path in polynomial time. The proposed approach gives intuitive correspondences for a variety of shapes and is robust in the presence of a wide range of visual transformations. The recognition rates on two distinct databases of 99 and 216 shapes each indicate highly successful within category matches (100 percent in top three matches), which render the framework potentially usable in a range of shape-based recognition applications.

On aligning curves

We present a novel approach to finding a correspondence (alignment) between two curves. The correspondence is based on a notion of an alignment curve which treats both curves symmetrically. We then define a similarity metric based on the alignment curve using two intrinsic properties of the curve, namely, length and curvature. The optimal correspondence is found by an efficient dynamic-programming method both for aligning pairs of curve segments and pairs of closed curves, and is effective in the presence of a variety of transformations of the curve. Finally, the correspondence is shown in application to handwritten character recognition, prototype formation, and object recognition, and is potentially useful in other applications such as registration and tracking.

A randomized linear-time algorithm to find minimum spanning trees

We present a randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered linear-time algorithm for verifying a minimum spanning tree. Our computational model is a unit-cost random-access machine with the restriction that the only operations allowed on edge weights are binary comparisons.

Faster Shortest-Path Algorithms for Planar Graphs

We give a linear-time algorithm for single-source shortest paths in planar graphs with nonnegative edge-lengths. Our algorithm also yields a linear-time algorithm for maximum flow in a planar graph with the source and sink on the same face. For the case where negative edge-lengths are allowed, we give an algorithm requiringO(n4/3log(nL)) time, whereLis the absolute value of the most negative length. This algorithm can be used to obtain similar bounds for computing a feasible flow in a planar network, for finding a perfect matching in a planar bipartite graph, and for finding a maximum flow in a planar graph when the source and sink are not on the same face. We also give parallel and dynamic versions of these algorithms.

When Trees Collide: An Approximation Algorithm for theGeneralized Steiner Problem on Networks

We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair

A nearly best-possible approximation algorithm for node-weighted Steiner trees

\{i,j\}

Computing the Edit-Distance between Unrooted Ordered Trees

of nodes, an edge-connectivity requirement

Excluded minors, network decomposition, and multicommodity flow

r_{ij}

Faster shortest-path algorithms for planar graphs

. The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within

Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut

2\lceil \log_2(r+1)\rceil