Finding a Happy Clique

 

Let $n_1,\ldots,n_k$ be the oriented nodes in $OV(\pi )$ in increasing left endpoint order (we ignore unoriented nodes at this stage). To locate a happy clique in $OV(\pi )$, the algorithm traverses the oriented nodes in $OV(\pi )$ according to this order. Let L(e) and R(e) be the left and right endpoints, respectively, of the interval corresponding to a node e in the realization of $OV(\pi )$. After traversing $n_1,\ldots,n_i$ for $1\le i\le k$, the algorithm maintains a happy clique C_i in the subgraph of $OV(\pi )$ induced by these vertices. Assume |C_i|=j, $j\le i$ and let $n_{i_1},\ldots,n_{i_j}$ be the vertices in C_iwhere $i_1 < i_2 <\ldots < i_j$. The vertices of C_i are maintained in a linked list ordered in increasing left endpoint order. If there exists an interval that contains all the intervals in C_i then the algorithm maintains a minimal such interval t_i. The clique C_i and the node t_i (if exists) satisfy the following invariant:

Invariant   $\;$
1) Every node $n_l\not\in C_i$, $l \le i$, such that L(n_i1) < L(n_l)must be adjacent to t_i.
2) Every node $n_l\not\in C_i$, L(n_l) < L(n_i1)that is adjacent to a node in C_i is either adjacent to an interval v_p such that R(n_p) < L(n_i1)or adjacent to t_i.

We prove the correctness of this invariant by induction: Initially $C_1=\{n_1\}$ and t₁ is undefined. If R(n_ij) < L(n_i+1)then C_i is guaranteed to be happy in $OV(\pi )$ (see figure 10.12(a)), therefore we need to focus only on cases with $L(n_{i+1}) \le R(n_{i_j})$. The induction step: We assume correctness up until i and show how to obtain C_i+1 and t_i+1 if $L(n_{i+1}) \le R(n_{i_j})$. We have to consider the following cases:
Case 1. The interval t_i is defined and R(t_i) <R(n_i+1). Continue with C_i+1 = C_i and t_i+1 = t_i. See figure 10.12(b).
Case 2. The interval t_i is not defined or $R(n_{i+1}) \le R(t_i)$.
a) R(n_ij) < R(n_i+1) and $L(n_{i+1}) \le R(n_{i_1})$. C_i+1 is obtained by adding n_i+1 to C_i and t_i+1 = t_i. See figure 10.12(c).
b) R(n_ij) < R(n_i+1) and L(n_i+1) > R(n_i1). The clique C_i+1consists of v_i+1 alone and t_i+1 = t_i. See figure 10.12(d).
c) R(n_i+1) < R(n_ij). As in the previous case $C_{i+1} = \{ n_{i+1} \}$. In this case t_i+1 is set to n_ij, the last interval in C_i. See figure 10.12(e).

  

Figure 10.12: The various cases of the algorithm for finding a happy clique. The topmost interval is always t_i. The three thick intervals comprise C_i. The dotted interval corresponds to n_i+1.

The fact that C_i is happy in the subgraph induced by $n_1,\ldots,n_i$ follows from this invariant. It is straightforward to see that the clique C_lthat the algorithm stops with, is happy.

The running time of the algorithm is proportional to the number of oriented nodes traversed since a constant amount of work is performed for each oriented node it encounters.