The simplest one is to execute find successor queries for all formula_1 entries, resulting in formula_33 initialization time. A better method is to check whether formula_20 entry in the finger table is still correct for the formula_35 entry. This will lead to formula_36. The best method is to initialize the finger table from its immediate neighbours and make some updates, which is formula_26. To ensure correct lookups, all successor pointers must be up to date. Therefore, a stabilization protocol is running periodically in the background which updates finger tables and successor pointers. The stabilization protocol works as follows:   With high probability, Chord contacts formula_26 nodes to find a successor in an formula_17-node network. Suppose node formula_21 wishes to find the successor of key formula_7. Let formula_42 be the predecessor of formula_7. We wish to find an upper bound for the number of steps it takes for a message to be routed from formula_21 to formula_42. Node formula_21 will examine its finger table and route the request to the closest predecessor of formula_7 that it has. Call this node formula_48. If formula_48 is the formula_20 entry in formula_21's finger table, then both formula_48 and formula_42 are at distances between formula_54 and formula_55 from formula_21 along the identifier circle. Hence, the distance between formula_48 and formula_42 along this circle is at most formula_54. Thus the distance from formula_48 to formula_42 is less than the distance from formula_21 to formula_48: the new distance to formula_42 is at most half the initial distance. This process of halving the remaining distance repeats itself, so after formula_65 steps, the distance remaining to formula_42 is at most formula_67; in particular, after formula_68 steps, the remaining distance is at most formula_69. Because nodes are distributed uniformly at random along the identifier circle, the expected number of nodes falling within an interval of this length is 1, and with high probability, there are fewer than formula_68 such nodes. Because the message always advances by at least one node, it takes at most formula_71 steps for a message to traverse this remaining distance. The total expected routing time is thus formula_26. If Chord keeps track of formula_73 predecessors/successors, then with high probability, if each node has probability of 1/4 of failing, find_successor (see below) and find_predecessor (see below) will return the correct nodes  Simply, the probability that all formula_13 nodes fail is formula_75, which is a low probability; so with high probability at least one of them is alive and the node will have the correct pointer.