三至五年级Discrete Mathematics计划中
Dots and lines: networks and puzzles
Networks of dots joined by lines: tracing figures without lifting the pencil, finding routes, and counting how many lines meet at each dot.
math.discrete.networks-puzzles
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前置知识
没有前置要求——从这里开始吧！
你将学会
- Decide by experiment which figures can be traced in one stroke without retracing a line.
- Count the line ends at each dot and notice the pattern behind traceable figures.
- Find shortest routes in a small road network with distances marked.
定义
- A network (graph) is a collection of dots (vertices) with lines (edges) joining some pairs of them.
- The degree of a dot is the number of line ends at it.
定理
- The degrees of all the dots add up to an even number: twice the number of lines.
- A connected figure can be traced in one stroke exactly when it has either no dots or exactly two dots of odd degree.
常见误解
- Thinking whether a figure can be traced depends on its overall shape rather than on the degrees of its dots.
- Believing the shortest route is always the one with the fewest roads.
通向
可选发现
- Algorithm design and analysis Finding shortest routes in huge networks, such as road maps, uses efficient algorithms such as Dijkstra's.