Almost Linear Time Algorithms for Some Problems on Dynamic Flow Networks

Motivated by evacuation planning, several problems regarding dynamic flow networks have been studied in recent years. A dynamic flow network consists of an undirected graph with positive edge lengths, positive edge capacities, and positive vertex weights. The road network in an area can be treated as a graph where the edge lengths are the distances along the roads and the vertex weights are the number of people at each site. An edge capacity limits the number of people that can enter the edge per unit time. In a dynamic flow network, when particular points on edges or vertices called sinks are given, all of the people are required to evacuate from the vertices to the sinks as quickly as possible. This chapter gives an overview of two of our recent results on the problem of locating multiple sinks in a dynamic flow path network such that the max/sum of evacuation times for all the people to sinks is minimized, and we focus on techniques that enable the problems to be solved in almost linear time.

Weighted Centrality and Retail Store Locations in Beijing, China: A Temporal Perspective from Dynamic Public Transport Flow Networks

The spatial relationship between transport networks and retail store locations is an important topic in studies related to commercial activities. Much effort has been made to study physical street networks, but they are seldom empirically discussed with considerations of transport flow networks from a temporal perspective. By using Beijing’s bus and subway smart card data (SCD) and point of interest (POI) data, this study examined the location patterns of various retail stores and their daily dynamic relationships with three weighted centrality indices in the networks of public transport flows: degree, betweenness, and closeness. The results indicate that most types of retail stores are highly correlated with weighted centrality indices. For the network constructed by total public transport flows in the week, supermarkets, convenience stores, electronics stores, and specialty stores had the highest weighted degree value. By contrast, building material stores and shopping malls had the weighted closeness and weighted betweenness values, respectively. From a temporal perspective, most retail types’ largest correlations on weekdays occurred during the after-work period of 19:00 to 21:00. On weekends, shopping malls and electronics stores changed their favorite periods to the daytime, while specialty stores favored the daytime on both weekdays and weekends. In general, the higher store type level of the shopping malls correlates more to weighted closeness or betweenness, and the lower-level store type of convenience stores correlates more to weighted degree. This study provides a temporal analysis that surpasses previous studies on street centrality and can help with urban commercial planning.