• 沒有找到結果。

Design of self-healing algorithm for ATM networks

N/A
N/A
Protected

Academic year: 2021

Share "Design of self-healing algorithm for ATM networks"

Copied!
6
0
0

加載中.... (立即查看全文)

全文

(1)

Design

of

Self-Healing Algorithm for

ATM

Networks

*

Wei-Bin Gao, Shi-Chung Chang and Chi-Tseng Chang

Department of Electrical Engineering

National Taiwan University

Taipei, Taiwan,

R.O.C.

email:[email protected]

.edu. tw

Abstract

A u t o m a t i c and rapid network service restoration (self-healing) w h e n failures occur i s of increasing im- portance f o r network operations and m a n a g e m e n t as network speed and bandwidth increase. In t h i s paper, w e s t u d y t h e self-healing problem of a n ATM network at t h e virtual p a t h (VP) configuration level. W e analyze t h e design issues o f self-healing algorithms, and devel- op a state t r a n s i t i o n and bandwidth division model of individual VPs a n d links. O n t o p of t h e model, we de- sign a restoration algorithm t h a t searches and activates alternate V P s f o r t h e failed V P s by combining a pre- p l a n m e t h o d and a double search m e t h o d . T h e preplan method activates preassigned backup V P s dynamically, while t h e double search m e t h o d searches f o r alternate VPs f r o m both t h e source a n d destination n o d e s of failed V P s . T h e double search m e t h o d is also applied t o reconstruction of t h e backup VP plan. S i m u l a t i o n s demonstrate t h a t o u r n e w algorithms o u t p e r f o r m ezist- ing preplanned and d y n a m i c self-healing algorithms in both restoration t i m e a n d restoration ratio.

I

Introduction

An ATM (asynchronous transfer mode) network [l] is a high-capacity optical fiber network, which trans- ports signals of various services such as voice, data, image, and video. As an ATM-based network offers a

high transimission capacity of integrated services, the failure of a network element (NE such as a link or a

node can cause a significant loss o

2

services t o users and

a loss of revenue to the network operating companies. It is estimated that if the network is disabled for one hour, up to $6,000,000 loss of revenue may occur in the trading and investment banking industries [2]. There- fore, automatic and rapid network service restoration (self-healing) when failures occur is of increasing im- portance for network operations and management.

In the literature, there have been a few ap- proaches for self-healing. In [3][4], a class of three- phase (route-search/acknowledgement/confirmation), distributed, flooding schemes are designed for self- healing in STM networks. The restoration is performed by the end nodes of the failed link. One of the nodes is called the sender node and the other is called the 'This work was supported in part by the National Science

Council of Republic of China, T a i w a n , under Grants NSCB5-

2213-E-002-045 and NSC86-2221-E002-016.

chooser node. When a failure occurs, the search mes- sage is initiated at the sender, flooded throughout the network, and finally terminated at the chooser. This algorithm is very simple and easy to implement, but its large number of restoration messages may waste the network resource or worsen the congestion situa- tion when failures occur. Yoshikai and Wu [5] proposed

a reliable protocol by extending the flooding algorithm to ATM networks with the consideration of self-healing control messages loss. To speed up the route search, a

double search method [6] is proposed that search mes- sages are initiated by both end nodes of a failed link.

In addition to the flooding algorithm-based sel€-healin schemes, many recent developements [7][8][9][1Of exploit the concept of the V P (virtual path) in ATM networks, and develop the class of pre- assignment or preplan methods. In a preplan method, each VP is assigned a backup V P in advance. When

a failure occurs, some backup VPs are activated to re- place the failed VPs. This technique simplifies the mes- sage transmission processes and reduces the number of restoration messages. Self-reconstruction techniques of backup VPs are also proposed.

To provide a prioritized restoration based on the quality of service (QoS) requirements of users, Chang and Huang [11] designed a scheme of prioritized backup VP assignment, where a high priority VP is assigned a backup VP and a reserved bandwidth while a low priority VP is only assigned a backup V P with zero bandwidth.

This paper focuses on designing a distributed algo- rithm for self-healing at VP configuration level under signle ATM network failure. It combines both the pre- plan and double search methods into a superior self- healing algorithm. The remainder of the paper is orga- nized as follows. Section E defines the self-healing prob- lem. A VP state transition model is then developed in Section

m.

Based on the model, a failed VP restoration algorithm and a backup VP reconstruction algorithm are proposed in Section

IV.

A simulation study in Sec- tion V demonstrates the superior time efficiency and restoration ratio of our self-healing algorithm. Finally, Section VI concludes the paper.

I1

Self-Healing Problem of

ATM

Net-

works

A virtual path (VP) in an ATM network is a pre- established path between two ATM network nodes, which may go through a few links and nodes with the same bandwidth provision. The topology and band-

(2)

width allocation of VPs in an ATM network is called

a V P configuration. In a VP configuration, a source- destination (S-D) pair may be sequentially connected via several VP links, [12].

There are two types of VPs : regular VPs, and back- up VPs [7]. A regular V P is a normal V P which has both a V P identifier (VPI) and a non-zero bandwidth allocation, whereas a backup V P has also a VPI but is allocated zero bandwidth. The objective of backup VPs is to provide alternate paths when regular VPs fail. Although a backup V P has the same terminal nodes with its corresponding regular VP, it goes through a

completely different set of links from the regular V P t o avoid a simultaneous failure of both. When a backup VP is activated, it can only use the residual bandwidth of individual physical links through which it passes. The bandwidth is called a captured bandwidth.

Since an ATM network supports high speed trans- port of multimedia services, its regular VPs and backup VPs are determined by the network management sys- tem based on QoS requirements of individual services and traffic demand distribution of the network. It is re- calculated periodically in order t o gain a n up-to-date optimality of resource allocation. After each compu- tation of VP configuration, the network management system downloads the V P configuration d a t a to each node (or switch).

When a link failure occurs, it must first be identified which VPs are on the failed link and all nodes on failed VPs should be notified about the failure. In order to recover the interrupted services on failed VPs, an al- ternate V P should be established to replace each failed VP. As an alternate V P may be prone to failure itself, a backup VP should be constructed for it as soon as the alternate VP is activated. Hence, there are three basic sub-problems for ATM network self-healing: fail- ure detection and diagnosis, alternate VP search and activation, and backup VP plan reconfiguration. Failure Detection and Diagnosis

In an ATM network, the availability of a VP is monitored and tested as part of the fault manage- ment functions of operation, administration, and main- tenance (OAM)[13][14]. ATM cells which transmit var- ious types of OAM information are called OAM cells. Among them, we concentrate on the type F4 OAM cells for V P failure notification, which can be gener- ated due t o a physical link failure or a node (switch) failure. Such an OAM cell should indicate the occur- rence of NE failure, the location of failure, and the VPs that are affected. We consider the alarm surveil- lance method for failure detection because it is the most time efficient among the three methods proposed by the ATM Forum [15].

A VP failure is reported in the ATM Layer with two types of cells: VP-AIS(A1arm Indication Signal) and VP-FERF Far End Report Failure). Upon receiving a

persistent

I

ailure indication from the Physical Layer, a detecting node will periodically issue VP-AIS cells to notify downstream nodes of the VP about connection unavailability until the fault is corrected. After receiv- ing a certain number of VP-AIS cells, the V P endpoint will begin t o send VP-FERF cells upstream to notify the source VP endpoint of the downstream failure.

Alternate VP Search and Activation

Once a failure is detected, the self-healing function should then find an alternate path for each failed V P t o recover the interrupted services. There are several issues of consideration:

physical connectivity: A self-healing system must search for an alternate V P that has the same ter- minal nodes with the failed V P but goes through

a completely different set of physical links. time constraint: There is a time limit for recov- ering the interrupted services, fot example, 2 sec- onds for a metropolitan LATA mesh network [16], which in turn poses a constraint on the number of V P links to search.

bandwidth capturing: An alternate V P should be allocated the same bandwidth as the bandwidth of the failed VP if possible to maintain a desirable QoS [17]. Routing tables in ATM switches should be modified accordingly.

Backup V P Plan Reconfiguration

After alternate VPs are found and activated, the in- terrupted services are recovered. A question then aris- es : "what should be the backup V P t o an alternate V P itself?". So the backup V P plan needs t o be re- configured based on the current V P configuration and physical link availability.

I11 Self-Healing

Algorithm

Design

To describe the dynamic system behavior of a V P configuration, we first define a state set

X

of seven states for a VP as follows. In the definition, both class- es regular VPs and activated (non-zero bandwidth) backup VPs are called active VPs. A V P that replaces a failed VP is called an alternate V P of the failed VP. State Set X E { X I , xz.

...,

27)

Model

x1 (activestatic state): an active VP with a backup VP;

x 2 (activedynamic state): an active VP with no back- up VP;

x 3 (r-invalid state): a V P in failure but with a n alter-

nate VP;

2 4 (invalid state): a VP in failure with no alternate

VP (including the backup VP in failure);

2 5 (backupinactive state): a backup V P with zero

bandwidth;

2 6 (backupsemiactive state): an activated backup V P

onto which no new calls can be added;

2 7 (inactive state): a repaired backup V P which is of zero bandwidth and backing up no other V P after repair.

There are nine events that may cause state transi- tions:

(3)

Event Set

E

E {el, e2,

...

,es}

el : the V P under discussion fails;

e2 : the V P under discussion is repaired;

e3 : the backup V P of the V P under discussion fails;

e4 : the backup V P of the VP under discussion is re- paired;

e5 : an alternate VP is found for the V P under discus- sion;

e6 : the backup V P is activated and captures a band- width after the regular V P under discussion fails;

e7 : the corresponding regular V P of the backup V P under discussion is repaired;

e8 : the V P under discussion is selected as a backup

e9 : the last call terminates on the backup V P under VP;

discussion.

State transitions axe shown in Fig.3.1. For exam- ple, consider a regular V P in an activestatic state and its backup V P in a backupinactive state. If now the regular VP fails, it transits to the invalid state, and its backup V P becomes activedynamic; If the fail- ure occurs on the backup VP, it then transits t o the invalid state, and its regular V P transmits t o be ac- tivedynamic. Since regular VPs and their bandwidth allocations are determined via optimization in the con- figuration phase, traffic provisioning should be through regular VPs whenever possible. So when a failed reg- ular VP is repaired, no new calls can be placed to its backup VP but the regular VP itself.

Link Bandwidth Division

In our model, the bandwidth of each link in the net- work is divided into three parts: normal bandwidth, re- served bandwidth and available bandwidth. A reserved bandwidth is allocated to backup VPs when activat- ed in order t o guarantee a given minimun restoration ratio. The normal bandwidth is the total allocated bandwidth of all VPs on a link. The available band- width equals to the bandwidth of a link bandwidth min- us both reserved bandwidth and normal bandwidth.

In the self-healing procedure, the alternate VP of

a failed VP may capture its bandwidth from different categories of link bandwidth based on the state of the failed VP. If a failed V P is activestatic, when activat- ing its backup VP, the bandwidth is to be captured from the available or the reserved bandwidth. In acti- vating an alternate VP of an activedynamic VP, the bandwidth is captured from the available bandwidth only.

IV Algorithm Design

On top of the model described in Section

ID,

we com- bine ideas of the preplan method and the dynamic dou- ble search method into a new self-healing algorithm in order t o achieve a short restoration time and a high

restoration ratio at the same time. When VP failures due to a single link failure occurs, if a failed V P is at

a n actives t at ic, activedynamic

,

or backupsemiact ive state, a restoration algorithm is designed t o find an al- ternate V P for it. If the state is backupinactive, we design a reconstruction algorithm t o construct a new backup VP. If the state is inactive, r-invalid, or invalid, there needs no actions because these failed VPs d o not affect the transmission traffic in the network and do not incur a revenue loss.

A regular V P can be either activestatic or ac-

tivedynamic before failure. The restoration algorithm adopts two methods of restoration based on the state of a regular VP when failure accurs: preplan method for an activestatic V P and double search method for an actived-ynamic VP.

IV.l

Preplan Method for Restoration

The preplan method preassigns a backup V P in advance for each regular V P and thus requires a short restoration time of searching for an alternate V P a t occurrence of a failure. Since the activestatic VPs have their individual backup VPs, when an activestatic VP fails, the preplan method first finds and activates its backup V P from the destination node to the source node. The required bandwidth should be captured by each node along the backup V P when activated. If any link en-route does not have enough available band- width to support the backup VP, the preplan method then fails, and the interrupted service on this failed V P is not recovered.

An algorithmic summary of the preplan method is described as follows.

PreDlan Method Algorithm

Step

Pl:

Activate the backup VP at the destination node.

Pl.1

Find the backup V P of the failed V P from the

node’s VPI mapping table;

P1.2 Change the failed VPI in the routing table t o its backup VPI;

P1.3 Send a restoration message t o the node one link upstream along the backup VP;

Step

P2:

Capture the required bandwidth and activate backup VP at an intermediate node.

P2.1 When a node, say n1, receives a restoration mes- sage from node n2, if the node is the source node for the VP to be restored, then go to step 3; P2.2 If (available

+

reserved) bandwidth

>

required

bandwidth on the link from node n1 to 7x2 that is

traversed by the backup VP,

then capture the required bandwidth from the available bandwidth first and then from the re- served bandwidth;

a release message is sent t o node to release all the bandwidths that have been captured from

n2 t o the destination node of the backup VP, and restoration of the failed V P stops;

endif.

P2.4

Activate one VP link upstream (towards the

source node) following the same procedure of Step P1 but a t the intermediate node.

(4)

Step P3: Complete restoration at the source node.

P3.1 When the source node receives a restoration mes- sage, it follows Steps P2.2 and P2.3;

P3.2 If it goes through 2.2, then go t o the reconstruc- tion algorithm steps since a backup VP is success- fully activated for restoring the failed VP and a

new backup VP is needed for the activated back- up VP itself.

Note that both restoration and release messages used in the method belong to the class of self-healing control (SHC) messages.

IV.2

Double Search Method

for

Restora-

tion

An activedynamic VP cannot be restored by the preplan method because it has no backup VP. Ideas of the distributed double search method by [6] are adopt- ed to search for and activate an alternate VP when an activedynamic VP fails. Both the source and destina- tion nodes of a failed activedynamic VP broadcast a

SHC search message for a VP link with the required bandwidth to all of their neighborhood nodes. Neigh- borhood nodes that can locally provide the required bandwidth will then relay the search t o their respec- tive neighborhood nodes, so on and so forth. When any node in the network receives two search messages originated from the source and destination nodes re- spectively, the node is defined to have a collision. The node combines the contents of the two colliding search messages t o construct a new SHC message called a re- sponse message and returns the response message along the coming paths of the two search messages t o source and destination nodes respectively. An alternate VP between the source and destination nodes is then es- tablished.

In searching for an alternate path by the double search method, there may be more than one colli- sions in the network because the searching messages are broadcasted, i.e., more than one alternate VPs may be identified. Information of these VPs are sent to the destination node by individual response messages. In this paper, the destination node takes the first re- sponse message arrival (and ignores the later ones) to construct the alternate VP. It then sends a completion message to complete the construction. If the source node of the failed VP does not receive a completion message in a limited amount of time, the restoration fails and stops.

Double Search Algorithm (Sketch) For the source n o d e :

Step S1; Broadcast a search message when failure of the actvie-dynamic VP is detected and start a

timer;

Step S2: If the timer value

>

a limit, then time is out, discard the restoration process, reset the timer and stop.

Step S3: Elseif a completion message is received,

change the failed V P I in the routing table to

its alternate VPI. For the destination node: Step

D1:

The same as Step S1;

Step 0 2 : The same as Step S2;

Step 0 3 : Elseif a response message is recevied, change the failed VPI in the routing table t o its alternate VPI.

Step

0 4 :

Send a completion message along the alter- nate VP.

Endif.

For any other node,say node

V,:

Step nl: The same as Step S2.

W h e n a search message i s received

Step n 2 : Assume that the search message is from node

V2.

If (available

+

reserved) bandwidth on the link between nodes V, and V, is larger than the required bandwidth, then reserve the required bandwidth over the link, add the link and node t o the path information of the search message, and ignore other search messages from the same origin (source or destination) node regarding the failed VP. Start a timer;

else ignore the search message. endif.

Step n3: Check if recevied search message from the source node collides with one from the destination node in their corresponding failure V P inform* tion.

Step n4: If not, generate and broadcast a search mes- sage to neighboring nodes.

Step n5: Else, synthesize a response message by com- bining the two colliding search messages and send the response messages t o nodes traverse by the two colliding search messages;

endif.

W h e n a response message i s received

Step n6: Update the routing table, capture appropri- ate bandwidth, and relay the response message t o the next node en-route.

W h e n a completion message is received

Step n7: Relay the completion message originated from the destination node to next node upstream along the alternate VP, check the routing table ac- cording t o the completion message, and reset the timer.

Both preplan and double search methods do not al- low partial restoration. Namely, a backup or alternate path must capture at least the same bandwidth as the failed VP. This of course may reduce the restoration ratio of our self-healing algorithm. However, consid- eration of partial restoration is expected t o increase the problem complexity and the decision time for self- healing.

IV.3

Reconstruction Algorithm

Reconstruction of backup VPs is needed on the fol- lowing three occasions: that backup VPs fail, that new VPs are established without backup VPs, and that net- work bandwidth allocations are changed due to failure restoration. Although the time requirement for finding

a backup VP is not so stringent as finding an alter- nate VP, it is still highly desirable t o be fast SO that each active VP has a backup VP to cope with the next failure. To reduce the search space, a limitation is set

(5)

through. If no backup V P can be found under the con- straint, then the reconstruction of a backup V P for the source-destination pair fails.

Our reconstruction algorithm design is also based on the double search method in order to keep the advan- tage of high restoration ratio and low restoration time. Once a need is identified for a backup V P between a

pair of source and destination nodes, a double search algorithmas described in Section

N

is used to establish

a backup VP except that SHC-help and confirmation messages are used for broadcasting and routing table setting instead, and that no bandwidth is assigned to the backup VP.

V

Simulation Results

In this section, we make up an examplary ATM net- work based on Taiwan’s ATM testbed network. The examplary network consists of 7 nodes and 15 bi- directional links, which has more links than reality in order to increase the options of alternate paths. TWO VP configurations are designed for the network, one with 20 VPs and the other with 30 VPs. Regular and backup VPs are generated in random, and route of a

regular VP is disjoint from the route of its backup VP. In both V P configurations, the reserved bandwidth per links is 10% of link capacity. In order to prevent all backup VPs from going through the few links, we control the bandwidth that can be used by backup VPs per link via a backup bandwidth parameter. This pa- rameter affects the probability of finding a backup VP and the distribution of VPs over the VP configuration. It is set to 5 times of the reserved bandwidth in our

simulation study.

Two types of failure situations are simulated: single link failure and a second link failure on the backup VP before the regular V P failure is repaired. A link fail- ure is randomly generated in each simulation run and Monte-Carlo simulation runs are conducted for each type of failure situation. We evaluate the restoration time and ratio performance of our self-healing algo- rithm and compare with self-healing methods of purely preplan method and purely double search (dynamic) method.

Figures 6.1-6.4 summarize the simulation results, where Figures 6.1 and 6.2 are for V P configuration 1

and Figures 6.3 and 6.4 are for configuration 2. Each figure indicates the average restoration ratio as a func- tion of the restoration time. From all these figures, we observe that given a restoration time less than 0.43 second, the preplan method is the best in restoration ratio because the backup VPs in the preplan method are preplanned via optimization. Our proposed algo- rithm stays quite close in average restoration ratio per-

formance to the preplan method. The dyanmic (double search) method searches for the alternate VPs in real time and is thus the worst in restoration ratio. When given a restoration time longer than 0.45 second, our proposed algorithm outperforms both preplan and dy- namic methods. The asymptotic performance between our proposed algorithm and the dynamic algorithm is close for both V P configurations. The restoration ratio of preplan method levels off quickly after 0.38 second. These results indicate that our proposed algorithm may perform well under a high VP density configuration and

when sequential link failures occur within a short du- ration.

VI

Conclusion

In this paper, based on the features of ATM net- work VP configuration, we developed a seven-state and bandwidth division model for self-healing algorithm de-

sign. A restoration algorithm which combines the pre- planned method and double search method is designed

t o deal with alternate VP search and activation for VPS at different states. A reconstruction algorithm was also developed based on the concept of double search method for backup VP reconstruction. Simulation re- sults demonstrated that our proposed self-healing algo- rithm indeed capture the features of both the preplan and the double search algorithms and thus achieved su-

perior performance in restoration ratio and restoration time. Futher evaluation and extension of the algorithm

to node and multiple failures are now underway.

References

[l] W. Stallings, ISDN and Broadband ISDN with

Frame Bealy and ATM, Prentice-Hall, Inc., New Jersey, 1995.

H. Komine, T. Chujo, T. Ogura, K. Miyazaki, and T. Soejima,”Adistributed Restoration Algorithm for Multiple-Link and Node-Failures of Transport Networks,” Proc. IEEE GLOBECOM ’90, pp.459- 463.

W.D. Grover,”The Self-Healing Network: A Fast Distributed Restoration Technique for Network- s using Digital Crossconnect Machines,”Proc.

GLOBECOM ’87, pp.28.2.1-28.2.6.

[4] S. Hasegawa, Y. Okanoue, T. Igawa, and H.

Sakauchi, ”Control Algorithms of SONET Inte- grated Self-Healing Networks,”IEEE J. Select Ar-

eas Commu., Vo1.12, No.1, Jan. 1994, pp. 110-119. N. Yoshikai, and T. Wu,”Control Protocol and Its Performance Analysis for Distributed ATM Vir- tual Path Self-Healing Network,”IEEE J. Select Areas Commu., Vo1.12, No.6, Aug. 1994, pp.1020- 1029.

H. Fujii, and N. Yoshikai,”Restoration Message Transfer Mechanism and Restoration Character- istics of Double-Search Self-Healing ATM Net- work,”IEEE J. Select Areas Commu., V01.12, No.1, January 1994.

R. Kawamura, K. Sato, and I. Tokizawa,”Self- Healing ATM Networks Based on Virtual Path Concept,”IEEE J. Select Areas Commu., V01.12,

No.1, January 1994.

R. Kawamura, and I. Tokizawa,”Self-Healing Vir- tual P a t h Architecture in ATM Networks,” IEEE

Commu. Mag., Sept. 1995.

[9] R. Kawamura, H. Hadama, and I. Tokiza- wa,”Implementaton of Self-Healing Function in ATM Networks Based on Virtual P a t h Concep-

[lo] J. Anderson, B. Doshi,

S.

Dravide, and P. Har- shavardhana, ”Fast Restoration of ATM Network- s,”IEEE J. Select Areas Commu., V01.12, No.1, [2] [3] [5] [SI [7] [8] t,”IEEE INFOCOM ’95, pp.303-311. pp.128-138, 1994.

(6)

[11] R. Chang, and C. Huang, 'Prioritized Restoration in ATM Networks", PTOC. IEEE ICOIN, 1997.

[12] K. Sato, and I. Tokizawa, "Flexible Asynchronous Transfer Mode Networks Utilizing Virtual Paths,"

Proc

ICC,

Apr. 1990, pp. 831-838.

[13] CCITT Study Group XVIII, Report R34, 143, June 1990.

[14] CCITT Study Group XVIII, Report R56, Annex

6, 80, Dec. 1990.

[Is]

D. McDysan, and D. Spohn, A T M Theory and Ap-

plication, McGraw-Hill, Inc., Taiwan, 1995. [16] Bellcore FA-NWT-001353, "Restoration of DCS

Mesh Networks with Dirtributed Control; Equip- ment Framework Generic Criteria," Dec. 1992. [17] B. Ryu, H. Ohsaki, M. Murata, and H. Miyahara,

"Design Algorithm for Virtual Path Based ATM Networks",

IEICE

Transactions on Commu., E79-

B, 2, pp.97-107, Feb, 1996.

( a ) stale transition model for regular VPs

W

(b) state transition niodel for backup VPs

Figure 3.1: State Transition Model

Figure 6.1: Self-Healing of First Failure for V P

Configuration 1

Figure 6.2: Self-Healing of Second Failure for VP Configuration 1

F

0.11

t

Figure 6.3: Self-Healing of First Failure for VP Configuration 2

y

1 0 0 200 aeu '00 sw

-

708

-

" u b n dnm

Figure 6.4: Self-Healing of Second Failure for VP Configuration 2

數據

Figure  6.1:  Self-Healing of  First  Failure for  V P   Configuration  1

參考文獻

相關文件

Other advantages of our ProjPSO algorithm over current methods are (1) our experience is that the time required to generate the optimal design is gen- erally a lot faster than many

Al atoms are larger than N atoms because as you trace the path between N and Al on the periodic table, you move down a column (atomic size increases) and then to the left across

Then, it is easy to see that there are 9 problems for which the iterative numbers of the algorithm using ψ α,θ,p in the case of θ = 1 and p = 3 are less than the one of the

Like the proximal point algorithm using D-function [5, 8], we under some mild assumptions es- tablish the global convergence of the algorithm expressed in terms of function values,

2-1 註冊為會員後您便有了個別的”my iF”帳戶。完成註冊後請點選左方 Register entry (直接登入 my iF 則直接進入下方畫面),即可選擇目前開放可供參賽的獎項,找到iF STUDENT

Elsewhere the difference between and this plain wave is, in virtue of equation (A13), of order of .Generally the best choice for x 1 ,x 2 are the points where V(x) has

• Definition: A max tree is a tree in which the key v alue in each node is no smaller (larger) than the k ey values in its children (if any). • Definition: A max heap is a

If necessary, you might like to guide students to read over the notes and discuss the roles and language required of a chairperson or secretary to prepare them for the activity9.