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1. Choose a hash function (at your own convenience) and 3 key values, and calculate their hash code. Choose the 3 key values so that 2 of them collide (but not all the 3). Although you may repeat a hash function already chosen, the key values should be distinct from all other postings.

2. The concept of Social Graph was introduced by M. Zuckerberg at the first Facebook F8 conference in 2007 when he introduced the Facebook platform to model relationships among internet users.
a) Discuss the importance of studying Social Graphs and how can they be used to better understand and process social relationships and
b) Using Web resources investigate the main concepts of Social Graphs. Choose one concept and discuss why it is relevant for Social Graphs and what social relation properties it measures.

3. DepthFirstSearch (DFS): parent node; discovery/finishing time; parenthesis theorem.

Requirements: implement/update specific methods for the DFS of a graph; for at least 2 graphs (1 being the provided one), show the DFS order of vertices in the graph, and for each node, specify its parent node in the search (the node from which the current node was reached). Moreover, display for each node the discovery and finishing time, to check that the Parenthesis Theorem holds true.

Approach: You may start from the DFT in the textbook and do the appropriate changes (or, find the alternative starting point with justification below and source, attached). For counting the dfs time, you should set a global counter (let’s name it time), which is set to 0 in the moment the search starts with the root vertex. At any moment the search starts with a new vertex (that is, just after entering a new dfs) the counter has to be incremented by one; also it has to be incremented by one when the search from the current node ends (just before exit a dfs). For each node, calculate also the discovery and finishing time; for doing this, you may use two arrays (let’s name them d[] for discovery and f[] for finishing). The discovery time of a given vertex v receives the value of the counter just when enter to the dfs for that vertex (that is d[v]=time, before incrementing time), and the finishing just before exit from that dfs (that is f[v]=time, before incrementing time).

Parentheses Theorem: In any depth first search, for any 2 vertices u and v, one of the following 3 conditions holds true:

*. The intervals [d[u], f[u]] and [d[v], f[v]] are completely disjoint;
*. The [d[u], f[u]] interval is completely contained within interval [d[v], f[v]], and u is a descendant of v in the dfs tree;
*. The [d[v], f[v]] interval is completely contained within interval [d[u], f[u]], and v is a descendant of u in the dfs tree;
Design and implement a driver to show the following (check for 2 graphs; 1 is provided, including the starting vertex):
- Display the dfs starting from a specified vertex;
- Display the parent node of each node in the dfs;
- Display the discovery/finishing time for each node in the graph;
- Show the Parentheses Theorem holds true, by mentioning the specific condition in each case (this may be manually calculated and added in the documentation, or shown within the application – design and implement a method to consider each pair and choose the appropriate case).

Input data: You should test your application and include the tests in your documentation for at least two graphs; one is mandatory to be this one provided below. It is represented in the G = (V, E) representation, where V is the vertices set, and E is the edges set. Please note that our graph is a directed one (that is edges have directions, thus, the presence of an (u, v) edge does not imply (v, u) is also present in the graph). Nevertheless, this has no impact on the algorithm and its implementation. The dfs should start with vertex 1.
V= {1, 2, 3, 4, 5, 6, 7}         
E= {{1, 2}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 5}, {4, 5}, {5, 1},{6, 4}, {6, 7}}

Deliverables: You should submit (1) all the source (.java) files, (2) a screenshot sample file (the output displayed while running your application to show the required functionalities) and (3) a documentation file. It should contain the design decisions, test plan, the output for the two runs, and the condition met by the   Parentheses Theorem in each case. The documentation should be no more than five pages in length and no less than two pages. The font size should be 12 point, the page margins one inch, and the paragraphs formatted with single spaced. Any figures, tables, equations, and references should be properly labeled and formatted using APA style.

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1. Choose a hash function (at your own convenience) and 3 key values, and calculate their hash code. Choose the 3 key values so that 2 of them collide (but not all the 3). Although you may repeat a hash function already chosen, the key values should be distinct from all other postings.

Hash function : h(x) = x % 3 + 5
Key values : 6 , 7, 9
h(6) = 5, h(7) = 9, h (9) = 5
we rehash with h(9) . h( h(9) ) = h(5) = 7...
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