Suppose a set of n objects is given, such that the size of the ith object satisfies that 0 < < 1. It is required to pack all the objects in a set of boxes of unit size. Each box can contain any subset of objects whose total size does not exceed 1. The heuristic first-fit takes each object and fits it into the first box that can hold it. Finally, let S be sum of the sizes of all objects: a. Show that the total number of boxes used by the first-fit heuristic is never greater than ⌈2S⌉. b. Show that the approximation factor of the first-fit heuristic is 2.

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Topological Sort

Topological sort is a sorting technique that sets up a hierarchy in a process. In this technique, when a directed network has nodes connected by arrows, the root node is placed before the successor node.