We revisit the shortest paths problem, considering the case where the input is a directed minor-free graph with negative arc lengths (but no negative-length cycles).In Lecture 14, we saw almost-linear-time algorithms for the case of planar and bounded-genus graphs. Currently, comparable bounds for minor-free graphs are not known. We shall discuss Goldberg's algorithm, a shortest-path algorithm for general graphs with integer lengths, whose running time depends logarithmically on the magnitude of the largest negative arc length. By exploiting separators (Lecture 6), it runs faster on minor-free graphs than on general graphs, but it still requires superlinear time.

Lectures:- Play ►
### Naming

00:47:59Hari BalakrishnanWe revisit the shortest paths problem, considering the case where the input is a directed minor-free graph with negative arc lengths (but no negative-length cycles).In Lecture 14, we saw almost-linear-time algorithms for the case of planar and bounded-genus graphs. Currently, comparable bounds for minor-free graphs are not known. We shall discuss Goldberg's algorithm, a shortest-path algorithm for general graphs with integer lengths, whose running time depends logarithmically on the magnitude of the largest negative arc length. By exploiting separators (Lecture 6), it runs faster on minor-free graphs than on general graphs, but it still requires superlinear time.

- Play ►
### Fault Isolation with Clients and Servers

00:50:28Hari BalakrishnanWe revisit the shortest paths problem, considering the case where the input is a directed minor-free graph with negative arc lengths (but no negative-length cycles).In Lecture 14, we saw almost-linear-time algorithms for the case of planar and bounded-genus graphs. Currently, comparable bounds for minor-free graphs are not known. We shall discuss Goldberg's algorithm, a shortest-path algorithm for general graphs with integer lengths, whose running time depends logarithmically on the magnitude of the largest negative arc length. By exploiting separators (Lecture 6), it runs faster on minor-free graphs than on general graphs, but it still requires superlinear time.

- Play ►
### Virtualization and Virtual Memory

00:49:47Hari Balakrishnan - Play ►
### Virtual Processors: Threads and Coordination

00:50:51Hari Balakrishnan - Play ►
### Performance

00:48:44Hari Balakrishnan - Play ►
### Introduction to Networks

00:50:29Hari Balakrishnan - Play ►
### Layering and Link Layer

00:47:35Hari Balakrishnan - Play ►
### Network Layer, Routing

00:50:55Hari Balakrishnan - Play ►
### End-to-end Layer

00:50:23Hari Balakrishnan - Play ►
### Distributed Naming

00:51:44Hari Balakrishnan - Play ►
### Reliability

00:49:08Hari Balakrishnan - Play ►
### Atomicity Concepts

00:50:29Hari Balakrishnan - Play ►
### Isolation

00:51:07Hari Balakrishnan - Play ►
### Transactions and Consistency

00:47:09Hari Balakrishnan - Play ►
### Multi-site Atomicity

00:50:49Hari Balakrishnan - Play ►
### Security Introduction

00:50:57Hari Balakrishnan - Play ►
### Authentication

00:51:44Hari Balakrishnan - Play ►
### Authorization and Confidentiality

00:39:32Hari Balakrishnan - Play ►
### Advanced Authentication

00:50:02Hari Balakrishnan - Play ►
### Complex, Trusted Systems

00:48:47Hal Abelson